Question

Differentiate the functions with respect to x.
$cos(sin\ x)$

Answer 1

Let f(x) = cos (sin x), u(x) = sin x, and v(t) = cos t
Then, (vou)(x) = v(u(x)) = v(sin x) = cos (sin x) = f(x)
Thus, f is a composite of two functions.
Put t = u(x) = sin x

Then, we obtain
$\frac {dv}{dt}$ = $\frac {d}{dt}$(cos t) = -sin t = -sin (sin x)
$\frac {dt}{dx}$ = $\frac {d}{dx}$(sin x) = cos x
Therefore by chain rule, $\frac {df}{dx}$ = $\frac {dv}{dt}$ . $\frac {dt}{dx}$ = - sin (sin x) . cos x= -cos x . sin (sin x)

Alternate method:

$\frac {d}{dx}$[cos (sin x)] = -sin (sin x) . $\frac {d}{dx}$(sin x)
= -sin (sin x) . cos x
= - cos x. sin (sin x)