Question

evaluate and simplify.
$\frac{d}{dx}sin(5sec x)$
$\frac{d}{dx}sin(5sec x)=square$

Explanation:

Step1: Apply chain - rule

Let $u = 5\sec x$. Then $\frac{d}{dx}\sin(5\sec x)=\frac{d}{du}\sin(u)\cdot\frac{du}{dx}$.
The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$. So, $\frac{d}{du}\sin(u)=\cos(u)$.

Step2: Find $\frac{du}{dx}$

Since $u = 5\sec x$, and the derivative of $\sec x$ is $\sec x\tan x$, then $\frac{du}{dx}=5\sec x\tan x$.

Step3: Substitute $u$ back and simplify

Substitute $u = 5\sec x$ into $\frac{d}{du}\sin(u)\cdot\frac{du}{dx}$. We get $\cos(5\sec x)\cdot5\sec x\tan x = 5\sec x\tan x\cos(5\sec x)$.

Answer:

$5\sec x\tan x\cos(5\sec x)$