Question

find $\frac{d^{2}}{dx^{2}}(sec x)$. $\frac{d^{2}}{dx^{2}}(sec x)=square$

Explanation:

Step1: Find first - order derivative

We know that the derivative of $\sec x$ with respect to $x$ is $\sec x\tan x$. So, $\frac{d}{dx}(\sec x)=\sec x\tan x$.

Step2: Find second - order derivative

Using the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = \sec x$ and $v=\tan x$. The derivative of $\sec x$ is $\sec x\tan x$ and the derivative of $\tan x$ is $\sec^{2}x$. Then $\frac{d^{2}}{dx^{2}}(\sec x)=\frac{d}{dx}(\sec x\tan x)=\sec x\tan x\cdot\tan x+\sec x\cdot\sec^{2}x=\sec x(\tan^{2}x + \sec^{2}x)$.

Answer:

$\sec x(\tan^{2}x+\sec^{2}x)$