# Derivative of tan cube x

Learn what is the derivative of tan cube x with formula. Also understand how to prove the derivative of tan^3x by different techniques.

Alan Walker-

Published on 2023-05-26

## Introduction to the Derivative of tan cube x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of tan^3x can be calculated by following the rules of differentiation. Or, we can directly find the derivative of tan^3x by applying the first principle of differentiation. In this article, you will learn what the derivative of tan cube x is and how to calculate the differentiation of tan cube x by using different approaches.

## What is the derivative of tan^3x?

The derivative tan^3x with respect to the variable x is given by the expression 3tan^2(x)sec^2(x), which is denoted by d/dx(tan^3(x)).

This represents the rate of change of the trigonometric function tan x, which is the ratio of the sine and cosine functions in a triangle. To be more precise, it is the ratio of the opposite side to the adjacent side. Therefore, the formula for tan x is given by sin x / cos x. Knowing this, we can differentiate tan cube x with ease by applying the chain rule of differentiation.

## Derivative of tan cube x formula

The formula for the derivative of the function tan^2(x) is equal to the product of the tangent and secant functions, which is given as:

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

This formula can be derived by applying the power rule of differentiation, which states that the derivative of a function raised to a power n is equal to n times the function raised to n-1, multiplied by the derivative of the function itself.

## How do you prove the tan cube x differentiation?

There are various rules of derivative to derive the derivative of the function tan^3(x). Two common methods are;

1. Chain Rule
2. Product rule

Each method provides a different way to compute the tan cube x differentiation. By using these methods, we can mathematically prove the formula for finding derivatives of tan^3.

## Differentiation of tan cube x by chain rule

To calculate the tan cube x formula derivative, 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 derivatives is defined as;

dy / dx = dy / du x du / dx

## Proof of derivative of tan^3x by chain rule

To prove the derivative of tan cube x by using dy/dt calculator, we start by assuming that,

y = u3 where u = tan x

By chain rule,

y = 3u2. du/dx

And,

du/dx = sec2 x

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

y = 3tan2 x.sec2 x

## Derivative of tan3 x 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 calculator 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 tangent cube by using product rule

To prove the derivative of tan cube x by using product rule of derivative, we have,

d/dx (tan3x) = d/dx(tan x.tan2 x)

d/dx (tan3x) = tan x. (tan2 x) + tan2 x. (tan x)

d/dx (tan3x) = tan x.( 2tan x.sec2 x)+ tan2 x.(sec2 x)

It can be written as;

d/dx (tan3x) = 3tan2 x.sec2 x

Hence the tan cube x differentiation is proved by using product rule.

## How to find the derivative of tan^3x with a calculator?

The easiest way to calculate the derivative of tan^3 is by using an online tool. You can find all derivative calculators on this website which assist you to solve such equations step by step.

1. Write the function as tan^3x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the tan cube x differentiation.
2. Now, select the variable by which you want to differentiate tan cube x. Here you have to choose x.
3. Select how many times you want to differentiate tan^3x. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button.

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

### What is the second derivative of tan?

The second derivative of tan x can be calculated by applying differentiation on the first derivative of tan x which is sec2x. Therefore, the second derivative of tan x is 2sec 2x tan x.

### What is the 1st and 2nd derivative?

The first derivative gives the slope of a tangent line in mathematics. Whereas the second derivative of a function tells whether the function is increasing or decreasing.

### Why do we use the 2nd derivative?

Since the first derivative calculates the rate of change of a function. The second derivative measures the concavity of the graph of the function. Therefore, it is used to determine that either the graph is concave up or down.