What is the derivative of ${{\tan }^{2}}x$?

Hint: We are asked to find the derivative of ${{\tan }^{2}}x$. We are going to find the derivative of ${{\tan }^{2}}x$ by using chain rule. In chain rule, we will assume $\tan x$ as “t” and then we are going to take the derivative with respect to “t”. Now, we are going to find the relation between “t” and “x” by equating “t” to $\tan x$ and then differentiating on both sides. Then write “t” in terms of x.

Complete step-by-step solution:
In the above problem, we have asked to find the derivative of ${{\tan }^{2}}x$. For that, we are going to assume $\tan x=t$. Now, substituting $\tan x$ as “t” in ${{\tan }^{2}}x$ we get,
${{t}^{2}}$
Taking derivative of the above expression with respect to “c” we get,
$2t\dfrac{dt}{dx}$ ………… (1)
The above derivative comes because we know that the differentiation of ${{t}^{n}}$ is given as:
$\dfrac{d{{t}^{n}}}{dt}=n{{t}^{n-1}}$
Now, we are going to differentiate on both the sides of $\tan x=t$ we get,
${{\sec }^{2}}xdx=dt$
Dividing $dx$ on both the sides of the above equation we get,
${{\sec }^{2}}x=\dfrac{dt}{dx}$
Using the above relation in eq. (1) we get,
$2t{{\sec }^{2}}x$
Substituting the value of “t” as $\tan x$ in the above equation we get,
$2\left( \tan x \right){{\sec }^{2}}x$
Hence, we have solved the derivative of ${{\tan }^{2}}x$ as $2\tan x{{\sec }^{2}}x$.

Note: To solve the above problem, you should know the derivative of $\tan x$ with respect to x and also how to differentiate ${{x}^{n}}$ with respect to x otherwise we cannot move forward in the above problem. Also, while finding the derivative of ${{\tan }^{2}}x$, you might forget to write 2 in the final answer so make sure you have written the number 2 in the final answer.