Introduction to the Derivative of tan^2x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of tan square x can be calculated by following the rules of differentiation. Or, we can directly find the differentiation of tan square x by applying the first principle of differentiation.

In this article, you will learn what the derivative of tan^2x is and how to calculate the tan square x differentiation by using different approaches.

What is the derivative of tan square x?

The derivative of tan^2x with respect to the variable x is equal to 2tan x.sec^2x. It is denoted by d/dx(tan2 x). It represents the rate of change of the trigonometric function tan x, which is the ratio of sine and cosine functions in a triangle. It is written as;
tan x = sin x / tan x

Derivative of tan squared formula

To find the derivative of tan squared x, we use the formula that involves the sine and cosine functions. The derivative tan square x formula is given as:

d/dx (tan^2 x) = 2tan x.sec^2 x

By differentiating tan squared x, we get 2 tan x multiplied by the secant squared x. This formula can be used to find the slope of the tangent line to the curve at any point on the graph. 

How do you prove the differentiation of tan square x?

There are different ways to derive the derivative of tan squared. Therefore, we can prove the derivative of tan2 x by using;

1. Chain Rule
2. Product rule

Each method provides a different way to compute the tan square x differentiation. By using these methods, we can mathematically prove the formula for finding the differential of tan square x. Also, these methods can be use to calculate the derivative of tan cube x.

Tan square x derivative by chain rule

To calculate the tan square x differentiation, we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivative is defined as;

dy / dx = dy / du x du / dx

Proof of derivative of tan^2x by chain rule

To prove the derivative of tan square x by using chain rule, we start by assuming that,

y = u2 where u = tan x

By using the chain rule calculator,

y = 2u. du/dx

And,

du/dx = sec2 x

now, using the value of u and du/dx in y, we have,

y= 2tan x.sec2 x

The derivative of tan3x can also be verified by using chain rule.

Tan square x differentiation by using product rule

The product rule is a formula used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule is:

d/dx(uv) = u(dv/dx) + (du/dx)v

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of tan^2x by using product rule

To differentiate tan square x by using product rule of derivatives, we have,

d/dx (tan2x) = d/dx(tan x.tan x)

d/dx (tan2x) = tan x. (tan x)+ tan x. (tan x)

d/dx (tan2x) = tan x.( sec2 x)+ tan x.(sec2 x)

It can be written as;

d/dx (tan2x) = 2tan x.sec2x

Hence the derivative of tan square x is proved by using product rule. You can use online product rule differentiation calculator for learning and practice.

How to find the derivative of tan^2x with a calculator?

The easiest way to calculate the differentiation of tan square x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as tan^2x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the tan square x derivative.
2. Now, select the variable by which you want to differentiate tan^2x. Here you have to choose x.
3. Select how many times you want to differentiate tan square x. In this step, you can choose 2 for solving second derivative, 3 for third derivative and so on.
4. Click on the calculate button. After this step, you will get the derivative of tan squared x within a few seconds.

After completing these steps, you will receive the tan square derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequently asked questions

Is sin2x differentiable?

A function is differentiable if it is continuous. Whereas a function is not differentiable if it is discontinuous at a point. The function sin2x is differentiable everywhere except at the point when it is zero.

What is the meaning of taking derivatives?

The derivative is the slope of the tangent line at any point on the curve. When you take a function derivative, you get another function that provides the slope of the original function.

Why is the slope of the tangent derivative?

The slope of the tangent line is the rate of change of the function at a point. As the definition of derivative is also a measure of rate of change, therefore the slope of tangent is derivative.