Question

What is the third derivative of $\tan x$?

Answer 1

In this problem we have to find the third derivative of the given trigonometric function, $\tan x$. Here the trigonometric function $\tan x$ should be differentiated three times one by one. Then we can get the solution, the third derivative of $\tan x$. While differentiating the trigonometric function $\tan x$we must use some differentiation conditions and formulas.

Complete step by step solution:
Here we have to find the third derivative of $\tan x$.
To get the third derivative of the trigonometric function $\tan x$we must differentiate it thrice and simplify it. Let us consider the trigonometric function $\tan x$as

$\Rightarrow y=\tan x$
Now differentiating it for the first time we get,
$\Rightarrow y=\tan x\dfrac{dy}{dx}$
By using the differentiation formula, differentiating $\tan x$we get,
$\Rightarrow y'={{\sec }^{2}}x$
We know that the first derivation of the trigonometric function $\tan x$ is over. Now differentiating the first derivative we get second derivative, so
$\Rightarrow y'={{\sec }^{2}}x\dfrac{d{{y}^{'}}}{dx}$
Here we can use the formula,$\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}$ and then differentiating rest of the $\sec x$ we get,
$\Rightarrow y''=2\sec x\sec x\tan x$ $\left[ \because \dfrac{d}{dx}\sec x=\sec x\tan x \right]$
Now multiplying the similar terms we get the second derivation,
$\Rightarrow y''=2{{\sec }^{2}}x\tan x$
We can see that the second derivation of the trigonometric function $\tan x$is over. Now differentiating the second derivative we get third derivative so,
$\Rightarrow y''=2{{\sec }^{2}}x\tan x\dfrac{dy''}{dx}$
Now we have to differentiate it though $uv$ method where the formula is ${{\left( uv \right)}^{'}}=u'v+uv'$ here,
$\Rightarrow u={{\sec }^{2}}x$
$\Rightarrow v=\tan x$
Now simplifying we can get the third derivative
$\Rightarrow y'''=2\left[ 2\sec x\sec x\tan x\tan x+{{\sec }^{2}}x{{\sec }^{2}}x \right]$
$\Rightarrow y'''=2\left[ 2{{\sec }^{2}}x{{\tan }^{2}}x+{{\sec }^{4}}x \right]$
**Therefore, the third derivative of the trigonometric function is $2\left[ 2{{\sec }^{2}}x{{\tan }^{2}}x+{{\sec }^{4}}x \right]$ or $4{{\sec }^{2}}x{{\tan }^{2}}x+2{{\sec }^{4}}x$.

Note:** In this solution we have found the third derivative of the trigonometric function $\tan x$. Here we have to remember some differentiation formulas. We have to simplify carefully while differentiating using the $uv$ method. Students make mistakes while simplifying it.