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

Chain Rule Calculator

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 Rule Calculator
Chain Rule Calculator

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.

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).