Question

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

Explanation:

Step1: Find the first - derivative

The derivative of $\tan x$ is $\sec^{2}x$. Using the constant - multiple rule $(cf(x))' = cf'(x)$ where $c = 7$ and $f(x)=\tan x$, we have $y'=\frac{d}{dx}(7\tan x)=7\sec^{2}x$.

Step2: Find the second - derivative

We use the chain rule to find the derivative of $y' = 7\sec^{2}x$. Let $u = \sec x$, then $y'=7u^{2}$. 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:

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