Question

example 8 if
f(x) = sin(cos(tan(x))),
then
f(x) = cos(cos(tan(x)))\frac{d}{dx}cos(tan(x))
= cos(cos(tan(x)))\boxed{}\frac{d}{dx}(tan(x))
= \boxed{}.
notice that we used the chain rule twice.

Explanation:

Step1: Differentiate cos(tan(x))

The derivative of $\cos(u)$ is $-\sin(u) \cdot u'$, so $\frac{d}{dx}\cos(\tan(x)) = -\sin(\tan(x)) \cdot \frac{d}{dx}(\tan(x))$. Fill the first blank with $-\sin(\tan(x))$.

Step2: Compute full derivative

Substitute the above into the expression: $f'(x) = \cos(\cos(\tan(x))) \cdot [-\sin(\tan(x))] \cdot \sec^2(x)$. Simplify to get the final form.

Answer:

First blank: $-\sin(\tan(x))$
Second blank: $\cos(\cos(\tan(x))) \cdot (-\sin(\tan(x))) \cdot \sec^2(x)$