Question

find y for the following function. y = 7 tan x y = □

Explanation:

Step1: Find first - derivative

The derivative of $\tan x$ is $\sec^{2}x$. Using the constant - multiple rule, if $y = 7\tan x$, then $y'=7\sec^{2}x$.

Step2: Find second - derivative

We use the chain rule to differentiate $y' = 7\sec^{2}x$. Let $u=\sec x$, so $y' = 7u^{2}$. The derivative of $y'$ with respect to $x$ is $\frac{dy'}{dx}=\frac{d(7u^{2})}{du}\cdot\frac{du}{dx}$. The derivative of $7u^{2}$ with respect to $u$ is $14u$, and the derivative of $\sec x$ with respect to $x$ is $\sec x\tan x$. Substituting $u = \sec x$ back in, we get $y''=14\sec x\cdot\sec x\tan x=14\sec^{2}x\tan x$.

Answer:

$14\sec^{2}x\tan x$