Table of Contents

**The derivative of tan x, sec x & tan x –** The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object’s velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the “instantaneous rate of change”, the ratio of the instantaneous change in the dependent variable to that of the independent variable.

**The derivative of tan x**

The derivative of tan x is sec2x.

To see why you’ll need to know a few results. First, you need to know that the derivative of sinx is cosx. Here’s a proof of that result from first principles

Once you know this, it also implies that the derivative of cosx is −sinx (which you’ll also need later). You need to know one more thing, which is the Quotient Rule for differentiation:

Once all those pieces are in place, the differentiation goes as follows:

**The derivative of sec x tan x**

Suppose we have y=f(x)⋅g(x)

Then, using Product Rule, y‘=f(x)⋅g‘(x)+f‘(x)⋅g(x)

In simple language, keep the first term as it is and differentiate the second term, then differentiate the first term and keep the second term as it is or vice-versa.

**Derivative of tanx^-1**

**Derivative of tanx^2**

In differential calculus, we use the **Chain Rule** when we have a composite function. It states that The derivative will be equal to the derivative of the outside function with respect to the inside, times the derivative of the inside function. Let’s see what that looks like mathematically

**Chain Rule**

f‘(g(x))⋅g‘(x)

Let’s say we have the composite function sin(5x). We know:

f(x)=sinx⇒f‘(x)=cosx

g(x)=5x⇒g‘(x)=5

So the derivative will be equal to

cos(5x)⋅5

=5cos(5x)

We just have to find our two functions, find their derivatives and input into the Chain Rule expression.

**using the chain rule for derivative of tanx^2**

Use the chain rule…

**What’s the derivative of SEC 2x?**

The **derivative of sec2 (x**) is 2sec**2 (x**) tan (x). The chain rule states that the **derivative** of f(g(x)) is equal to f ‘ (g(x)) ⋅ g ‘ (x)

**What is the differentiation of theta?**

**derivative**of the variable you are taking with respect to the variable. For example, if you are taking the

**derivative**of θ with respect to θ, you would get one. Generally, however, the

**derivative**is taken with respect to x, because x is the most commonly used variable.

**What is Sec x?**

**secX**is just the ratio of the length of a hypotenuse to the length of an adjacent side.

**What are the derivatives of the 6 trig functions?**

**Derivatives**of

**Trigonometric Functions**. The basic

**trigonometric functions**include the following

**6 functions**: sine (sinx), cosine (cosx), tangent (tanx), cotangent (cotx), secant (secx) and cosecant (cscx). All these

**functions**are continuous and differentiable in their domains.

**Is TANX continuous?**

**tan x**is not

**continuous**but is

**continuous**on for example the interval −π/2 <x<π/2. It has infinitely many points of discontinuity, at ±π/2, ±3π/2, etc.; all are infinite discontinuities.

**How do you find a derivative?**

**Basically, we can compute the derivative of f(x) using the limit definition of derivatives with the following steps:**

- Find f(x + h).
- Plug f(x + h), f(x), and h into the limit definition of a derivative.
- Simplify the difference quotient.
- Take the limit, as h approaches 0, of the simplified difference quotient.

**Is Tan Sin over COS?**

**cosine**. … The

**tangent**of x is defined to be its sine divided by its

**cosine**:

**tan**x =

**sin**x

**cos**x . The

**cotangent**of x is defined to be the

**cosine**of x divided by the sine of x: cot x =

**cos**x

**sin**x .

**What is COTX?**

**What is ‘cotx**‘?

**cot**is a short way to write ‘cotangent’. This is the reciprocal of the trigonometric function ‘tangent’ or tan(

**x**). Therefore,

**cot**(

**x**) can be simplified to 1/tan(

**x**). Using trigonometric rules, an alternative way to write 1/tan(

**x**) is cos(

**x**)/sin(

**x**).

**For what numbers is tan not defined?**

The tangent function, **tan**(x) is undefined when x = (π/2) + πk, where k is any integer.