# Chain Rule Calculator

Chain Rule Calculator is a free online tool that displays the derivative value of a given function. With STUDYQUERIES’ online chain rule calculator tool, you can quickly calculate derivatives and indefinite integrals.

## How to Use the Chain Rule Calculator?

To use the chain rule calculator, follow these steps:

• Step 1: Enter the function into the input field
• Step 2: Click the “Submit” button to get the derivative value
• Step 3: In the new window, the derivatives and the indefinite integral for the given function will be displayed

## What Is Chain Rule?

The chain rule is a method of finding the derivative of composite functions or functions made by combining one or more functions. An example of one of these types of functions is $$f(x)=(1+x)^2$$ which is formed by taking the function $$1+x$$ and plugging it into the function $$x^2$$. The chain rule can be applied to a surprising number of functions that can be thought of as composites.

Chain rules are needed for the following problems. Chain rules distinguish compositions of functions. In the following discussion and solutions, the derivative of the function h(x) will be denoted by $$D(h(x))\ or\ h'(x)$$ Most problems are average. There are a few difficult problems. Formally, the chain rule states that

$$D\left[f \left\{g \left(x \right)\right\}\right]=f’\left\{\left(g \right)x \right\}g’\left(x \right)$$

The chain rule is rarely applied in a formal manner when addressing specific problems. In most cases, we use intuition instead. Sometimes it is easier to consider the functions f and g as “layers” of a problem. The outer layer is function f, and the inner layer is function g.

Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged $$(the\ term\ f’\left\{g \left(x \right) \right\}$$, then differentiate the inner layer $$(the\ term\ g’\left(x \right) )$$ This process will become clearer as you do the problems. Final answers are usually given in the simplest form.

### Step By Step Guide To Apply The Chain Rule For Finding A Derivative Of Composite Functions

Let’s apply the chain rule by differentiating $$h\left(x \right) = (5-6x)^5$$

$$h\left(x \right) = (5-6x)^5$$

$$g\left(x \right) = 5-6x \longrightarrow\ Inner\ Function$$

$$f\left(x \right) = x^5 \longrightarrow\ Outer\ Function$$

Because h is composite, we can differentiate it using the chain rule:

$$\frac{d}{dx}\left[f\left\{g\left(x \right) \right\} \right]=f’\left\{\left(g \right)x \right\}.g’\left(x \right)$$

The rule states that the derivative of the composite function is the inner function g within the derivative of the outer function f’, multiplied by the derivative of the inner function g’.

Let’s find the derivatives of the inner and outer functions before applying the rule:

$$g’\left(x \right) = -6$$

$$f’\left(x \right) = 5x^4$$

Now let’s apply the chain rule:

$$\frac{d}{dx}\left[f\left\{g\left(x \right) \right\} \right]=f’\left\{\left(g \right)x \right\}.g’\left(x \right)$$

$$=5\left(5-6x \right)^4.-6$$

$$=-30\left(5-6x \right)^4$$

Example: Find the derivative of $$f(x)=(3x+1)^5$$

Solution: As seen in this example, there is a function 3x+1 that has been raised to the 5th power. Therefore, there are two pieces: 3x+1 (the inside function) and 5x+1 (the outside function). You know by the power rule, that the derivative of $$\frac{d}{dx}x^n=nx^\left(n-1 \right)$$

So, $$\frac{d}{dx}x^5=5x^4$$

Cover up that 3x+1 for a moment, and pretend it is an x. You will, however, have to pay a penalty. Because it wasn’t just an x, you’ll need to multiply by the derivative of 3x+1.

$$\frac{d}{dx}f\left(x \right)=\frac{d}{dx}\left(3x+1 \right)^5$$

$$=5\left(3x+1 \right)^4\frac{d}{dx}\left(3x+1 \right)$$

$$=5\left(3x+1 \right)^4\left(3 \right)$$

$$=15\left(3x+1 \right)^4$$

Example: Find the derivative of $$f\left(x \right)=ln\left(x^2-1 \right)$$

Solution: The same idea applies here. Normally, you would define the derivative as 1/x if the ln(x) was all there was. However, something else existed (the inside function). Cover it up and take the derivative anyway. When you are done, don’t forget to multiply by the derivative of the inside function.

Here is another example of finding the derivative using the chain rule. The derivative of ln of x squared minus 1 is 1 over x squared minus 1 times the derivative of x squared minus 1.

$$\frac{d}{dx}f\left(x \right)=\frac{d}{dx}ln\left(x^2-1 \right)$$

$$=\frac{1}{\left(x^2-1 \right)}\frac{d}{dx}\left(x^2-1 \right)$$

$$=\frac{1}{\left(x^2-1 \right)}\left(2x \right)$$

$$=\frac{2x}{\left(x^2-1 \right)}$$

Now you can simplify to get the final answer:

$$f′(x)=\frac{2x}{\left(x^2-1 \right)}$$

### Common Mistakes To Identify Composite Functions

A function is composite if you can write it as $$f\left\{g\left(x \right) \right\}$$ In other words, it is a function inside another function or a function itself.

For example,$$cos\left(x^2 \right)$$  is composite, because if we let $$f\left(x \right)= cos\left(x \right)$$ and $$g\left(x \right)= \left(x^2 \right)$$, then $$cos\left(x^2 \right) = f\left\{g\left(x \right) \right\}$$

g is the function within outer f, therefore we call g the “inner” function, and f the “outer” function.

​On the other hand,$$cos\left(x \right).x^2$$  is not a composite function. It is the product of $$f(x)= cos\left(x \right)\ and\ g(x)= x^2$$ however neither of the functions is within the other.

Not recognizing whether a function is composite or not

A composite function can usually only be differentiated by using the chain rule. We will not be able to differentiate correctly if we don’t recognize that a function is composite and the chain rule needs to be applied.

In contrast, if the chain rule is applied to a non-composite function, the derivative will be incorrect.

Especially with transcendental functions (e.g., trigonometric and logarithmic functions), students often confuse compositions like $$ln\ sin\left(x \right)$$ with products like $$ln\left(x \right)sin\left(x \right)$$

Wrong identification of the inner and outer function

Even when a student recognizes that a function is composite, they might get the inner and outer functions wrong. The derivative will inevitably be incorrect.

For example, in the composite function $$sin^2\left(x \right)$$ the outer function is $$x^2$$ and the inner function is $$sin\left(x \right)$$ Students are often confused by this sort of function and think that $$sin\left(x \right)$$ is the outer function.

## Chain Rule Derivative Calculator:

The Chain Rule Derivative Calculator is a tool that helps compute the derivative of a composite function using the chain rule. The chain rule is a fundamental concept in calculus used to find the derivative of a composition of functions. It states that if a function is composed of two or more functions, then the derivative of the composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

For example, let’s consider the function f(x) = sin(3x^2). To find its derivative using the chain rule, we can first identify the outer function as sin(x) and the inner function as 3x^2. Applying the chain rule, the derivative of f(x) with respect to x would be:

f'(x) = cos(3x^2) * (6x)

Using the Chain Rule Derivative Calculator, you can input the function expression, and it will provide the derivative with the steps involved in the calculation.

## Multivariable Chain Rule Calculator:

The Multivariable Chain Rule Calculator is designed to find the derivative of a function with multiple variables using the chain rule. In calculus, when dealing with functions that depend on multiple variables, the chain rule extends to the partial derivatives of each variable.

For example, consider the function f(x, y) = sin(x^2 + y^2). To find its partial derivative with respect to x, we would use the multivariable chain rule. The chain rule states that the partial derivative of the composite function is equal to the partial derivative of the outer function multiplied by the partial derivative of the inner function.

Using the Multivariable Chain Rule Calculator, you can input the function expression and the variables for which you want to compute the partial derivatives. The calculator will provide the partial derivatives with respect to each variable, taking into account the chain rule.

## Partial Derivative Chain Rule Calculator:

The Partial Derivative Chain Rule Calculator is specifically designed to calculate partial derivatives of composite functions. In multivariable calculus, the chain rule extends to partial derivatives, allowing us to find the derivative of a composition of functions with respect to a specific variable.

For example, let’s consider the function f(x, y) = sin(3x^2 + y). To find the partial derivative of f with respect to x, we apply the partial derivative chain rule. The result would be:

∂f/∂x = cos(3x^2 + y) * (6x)

Using the Partial Derivative Chain Rule Calculator, you can input the function expression and the variable for which you want to calculate the partial derivative. The calculator will provide the partial derivative with respect to the specified variable, considering the chain rule.

## Chain Rule Calculator With Steps:

The Chain Rule Calculator With Steps is a tool that not only calculates the derivative using the chain rule but also provides a step-by-step breakdown of the calculation. It helps users understand the process of applying the chain rule to find derivatives.

For example, if you input the function f(x) = (x^2 + 1)^3 into the Chain Rule Calculator With Steps, it will provide the derivative and explain each step involved in the calculation. The output might show the simplification of the function, identification of the inner and outer functions, and the application of the chain rule.

This calculator is helpful for students or anyone learning calculus as it provides a detailed explanation of the derivative calculation using the chain rule.

Chain Rule Partial Derivative Calculator:

The Chain Rule Partial Derivative Calculator is a tool specifically designed to calculate partial derivatives using the chain rule. It is useful when dealing with functions that depend on multiple variables and finding the derivative with respect to a specific variable.

For example, suppose we have

a function f(x, y) = sin(3x^2 + y). To find the partial derivative ∂f/∂x using the chain rule, the calculator would provide the step-by-step calculation, taking into account the composition of functions and the chain rule.

This calculator is particularly helpful in contexts where functions involve multiple variables and require finding partial derivatives while considering the chain rule.

## Chain Rule Calculator With Steps Free:

The Chain Rule Calculator With Steps Free is a free tool that combines the functionality of the Chain Rule Calculator and the step-by-step explanation feature. It allows users to calculate derivatives using the chain rule and provides a detailed breakdown of the calculation process, all without any cost.

This calculator is useful for students, educators, or anyone who wants to understand and practice the application of the chain rule in finding derivatives. It eliminates the need for manual calculations and provides a comprehensive explanation of each step involved.

## Dz/Dt Chain Rule Calculator:

The Dz/Dt Chain Rule Calculator is a specialized tool designed to calculate the derivative dz/dt using the chain rule. This calculator is particularly useful when dealing with functions that involve multiple variables and the derivative with respect to a specific variable (in this case, t).

For example, suppose we have a function z(x, y, t) = x^2 * sin(yt). To find the derivative dz/dt using the chain rule, the calculator would provide the step-by-step calculation, considering the composition of functions and the chain rule.

This calculator is valuable in fields such as physics, engineering, or any domain where functions have dependencies on multiple variables and require finding the derivative with respect to a specific variable.

## Diff Chain Rule Calculator:

The Diff Chain Rule Calculator is a tool that helps calculate derivatives using the chain rule for differentiation. It simplifies the process of finding the derivative of composite functions, where each function may have its own derivative.

For example, if you have a composite function f(g(x)), the Diff Chain Rule Calculator allows you to input the function expression and find its derivative. The calculator will break down the calculation into steps, considering the chain rule and differentiating each function along the way.

This calculator is useful for anyone working with complex functions and needing to find their derivatives efficiently using the chain rule. It eliminates the need for manual calculations and provides quick and accurate results.

## Multivariable Chain Rule Calculator:

The Multivariable Chain Rule Calculator is a tool specifically designed to find derivatives of functions with multiple variables using the chain rule. It accounts for the different partial derivatives with respect to each variable and provides the overall derivative.

For example, suppose we have a function f(x, y, z) = x^2 + 2yz. To find the derivative using the multivariable chain rule, the calculator would provide the step-by-step calculation, taking into account the partial derivatives of each variable and their respective chain rules.

This calculator is valuable in fields such as economics, physics, or any domain involving functions with multiple variables and the need to calculate derivatives accurately using the chain rule.

## Conclusion

You must learn to see functions differently because composite functions come in all kinds of forms. When trying to determine whether the chain rule makes sense for a particular problem, look for functions that have more complicated xs. Some examples are $$e^\left(5x \right), cos\left(9x^2 \right), and \frac{1}{x^2−2x+1}$$ All of these are composite functions and for each of these, the chain rule would be the best approach to finding the derivative.

## FAQs

How do you do the chain rule step by step?

• Step 1: Find the inner function and rewrite the outer function, replacing the inner function with u.
• Step 2: Calculate the derivative of both functions.
• Step 3: Incorporate the derivatives and the original expression for the variable u into the Chain Rule and simplify.

What is the chain rule in words?

The chain rule states that. (f(g(x)))’ = f ‘ (g(x)) · g ‘ (x). When we state the chain rule using words instead of symbols, it says this: to determine the derivative of f(g(x)), identify the outside and inside functions.

What is the chain rule for integration?

Anyway, the chain rule says if you take the derivative with respect to x of f(g(x)) you get f'(g(x))*g'(x). That means if you have a function in that form, you can take the integral of it to look like f(g(x)). The process of doing this is traditionally u-substitution.

Can you explain how the chain rule works in real life?

In the real world, we can also use the Chain Rule to estimate rates of change. With the Chain Rule, we can see how variables like time, speed, distance, volume, and weight are interconnected.

Where can we use the chain rule?

We use the chain rule when differentiating a ‘function of a function’, like f(g(x)) in general. We use the product rule when differentiating two functions multiplied together, like f(x)g(x) in general. Take an example, f(x) = sin(3x).

How To Type The Chain Rule At The Ti-89 Calculator?

To type the chain rule on a TI-89 calculator, you can use the following steps:

Step 1: Enter the outer function: Press the appropriate keys to enter the outer function in the calculator. For example, if the outer function is sin(x^2), you would enter “sin(x^2)”.

Step 2: Multiply by the derivative of the inner function: After entering the outer function, multiply it by the derivative of the inner function. To differentiate the inner function, use the “d/dx” operator followed by the inner function. For example, if the inner function is x^2, you would multiply the outer function by “d/dx(x^2)”.

Step 3: Simplify if needed: The calculator should automatically simplify the expression if possible. If not, you can simplify it manually using the appropriate calculator functions.

How To Do Chain Rule Given X And Y On A Calculator?

To apply the chain rule given x and y on a calculator, follow these steps:

Step 1: Identify the outer and inner functions: Determine the composition of functions in the given expression and identify the outer and inner functions. For example, if you have f(x, y) = sin(x^2 + y), the outer function is sin(z) where z = x^2 + y, and the inner function is z = x^2 + y.

Step 2: Calculate the partial derivatives: Differentiate the inner function with respect to both x and y separately. Use the appropriate partial derivative notation (∂z/∂x and ∂z/∂y) to calculate the partial derivatives.

Step 3: Apply the chain rule: Multiply the partial derivatives from step 2 by the derivative of the outer function. In this example, the chain rule would be: ∂f/∂x = cos(z) * (∂z/∂x) and ∂f/∂y = cos(z) * (∂z/∂y).

How To Solve Cot^3(4X+1) Using The Chain Rule Calculator?

To solve the expression cot^3(4x+1) using the chain rule calculator, follow these steps:

Step 1: Identify the outer and inner functions: In this case, the outer function is ( )^3, and the inner function is 4x + 1.

Step 2: Calculate the derivative of the inner function: Differentiate the inner function with respect to x. The derivative of 4x + 1 is simply 4.

Step 3: Apply the chain rule: Multiply the derivative of the outer function (3 cot^2(4x+1)) by the derivative of the inner function (4). The result will be the derivative of the entire expression.

Using the chain rule calculator, input the expression cot^3(4x+1), and it will provide the derivative with the steps involved in the calculation.

How To Use The Chain Rule With A Calculator?

To use the chain rule with a calculator, follow these general steps:

Step 1: Identify the outer and inner functions: Identify the composition of functions in the given expression and determine the outer and inner functions.

Step 2: Calculate the derivative of the inner function: Differentiate the inner function with respect to the variable involved. This may require using the appropriate derivative rules for the specific function.

Step 3: Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function. This can be done by multiplying the expressions or functions involved.

Step 4: Simplify if needed: Simplify the resulting expression, if possible, using algebraic simplification techniques or calculator functions.

The specific steps may vary depending on the calculator you are using. It is recommended to consult the calculator’s user manual or refer to the specific chain rule calculator’s instructions for precise guidance.

How To Do Chain Rule On A Calculator?

To do the chain rule on a calculator, follow these general steps:

Step 1: Identify the outer and inner functions: Identify the composition of functions in the given expression and determine the outer and inner functions.

Step 2: Calculate the derivative of the inner function: Differentiate the inner function with respect to the variable involved. Use the appropriate derivative rules or operator provided by the calculator.

Step 3: Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function. This can be done by multiplying the expressions or functions involved.

Step 4: Simplify if needed: Simplify the resulting expression, if possible, using algebraic simplification techniques or calculator functions.

The specific steps and buttons to use may vary depending on the calculator model. It is recommended to consult the calculator’s user manual or refer to the specific calculator’s instructions for precise guidance.

How Do You Calculate Chain Rule?

To calculate the chain rule, follow these steps:

Step 1: Identify the outer and inner functions: Identify the composition of functions in the given expression and determine the outer and inner functions.

Step 2: Calculate the derivative of the inner function: Differentiate the inner function with respect to the variable involved. Apply the appropriate derivative rules for the specific function.

Step 3: Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function. This can be done by multiplying the expressions or functions involved.

Step 4: Simplify if needed: Simplify the resulting expression, if possible, using algebraic simplification techniques or mathematical properties.

The chain rule is a fundamental concept in calculus and is used to find derivatives of composite functions. By following the steps above and applying the chain rule correctly, you can calculate the derivative of a composition of functions.

What Is The Chain Rule Class 12?

In the context of education, specifically at the class 12 level, the chain rule refers to a fundamental concept taught in calculus. Class 12 typically covers advanced topics in mathematics, including calculus.

The chain rule is a rule used to find the derivative of a composite function. It allows you to differentiate functions that are composed of other functions. The chain rule is important in calculus as it enables the calculation of derivatives for a wide range of functions.

In class 12, students learn about differentiation and various derivative rules, including the chain rule. They practice applying the chain rule to find derivatives of composite functions and solve related problems.

The chain rule is a key concept in calculus and serves as a foundation for more advanced topics in mathematics, physics, engineering, and other fields.

What Is The Formula For Chain Rule With 3 Functions?

The formula for the chain rule with three functions can be stated as follows:

If y = f(g(h(x))), where f, g, and h are differentiable functions, then the derivative of y with respect to x can be calculated using the chain rule as:

dy/dx = (df/dg) * (dg/dh) * (dh/dx)

In this formula, (df/dg) represents the derivative of the outermost function f with respect to the intermediate function g, (dg/dh) represents the derivative of the intermediate function g with respect to the innermost function h, and (dh/dx) represents the derivative of the innermost function h with respect to the variable x.

By applying the chain rule formula for three functions, you can find the derivative of the composite function y with respect to x.

What Is The Chain Rule Formula Newton?

The chain rule is a concept in calculus that was not introduced by Isaac Newton but rather by Gottfried Wilhelm Leibniz, another prominent mathematician of the same era. Newton and Leibniz independently developed calculus, and both made significant contributions to the field.

The chain rule formula, as used in calculus today, was formalized by Leibniz and is attributed to him. The chain rule allows the calculation of derivatives of composite functions, which is a fundamental concept in calculus.

While Newton and Leibniz both contributed to the development of calculus, it is important to credit Leibniz specifically for the formulation of the chain rule as we know it today.