Question

1. if ( f(x)=sin (cos (\tan x)) ), find ( f^{prime}(x) )

Explanation:

Step1: Apply the chain rule

Let \(u = \cos(\tan x)\), then \(f(x)=\sin(u)\). By the chain rule \(\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}\).
We know that \(\frac{d}{du}(\sin u)=\cos u\).

Step2: Differentiate \(u = \cos(\tan x)\)

Let \(v=\tan x\), then \(u = \cos(v)\). By the chain rule \(\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}\).
We know that \(\frac{d}{dv}(\cos v)=-\sin v\) and \(\frac{d}{dx}(\tan x)=\sec^{2}x\).

Step3: Substitute back

Substitute \(u = \cos(\tan x)\) and \(v = \tan x\) into the expressions.
\(\frac{df}{dx}=\cos(\cos(\tan x))\cdot(-\sin(\tan x))\cdot\sec^{2}x\)

Answer:

\(-\sec^{2}x\sin(\tan x)\cos(\cos(\tan x))\)