Question

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

Answer 1

In this problem, we have to find the derivative of ${{\tan }^{2}}y$. Here we can use the chain rule as we have two functions in the given problem. Here we have an inside function $\tan y$ and the outside function ${{\tan }^{2}}y$, where we have to find the derivative of the outside function first and then we have to derive the inside function and then multiply them to get the final answer.

Complete step-by-step solution:
Here we have to find the derivative of ${{\tan }^{2}}y$.
We can now use the differentiation formulas to find the derivative of the given problem.
Here we have to use the chain rule as we have two functions.
We have an inside function $\tan y$ and the outside function ${{\tan }^{2}}y$, where we have to find the derivative of the outside function first and then we have to derive the inside function
We know that, $\dfrac{d}{dx}{{\tan }^{2}}u=2{{\tan }^{1}}u\dfrac{du}{dx}$
We can apply this to the outside function ${{\tan }^{2}}y$, we get
$\Rightarrow 2\tan y$
We know that $\dfrac{d}{dx}\tan y={{\sec }^{2}}y\dfrac{du}{dx}$
We can now differentiate the inside function $\tan y$, we get
$\Rightarrow 2\tan y{{\sec }^{2}}y$
Therefore, the derivative of ${{\tan }^{2}}y$ is $2\tan y{{\sec }^{2}}y$.

Note: We should always remember the chain rule, as we have two functions, we should take them as inside function and outside function. We can then find the derivatives for both inside and the outside functions and multiply them to get the final answer. We should also remember the formulas of derivation as we differentiate $\tan y$, we get ${{\sec }^{2}}y$.