Composite, Implicit, and Inverse Functions

How does one approach the differentiation of $\cos(e^{\tan^{-1}(\sqrt{x})})$ using chain rule while avoiding common mistakes associated with nested functions?

If $f(x) = g(h(x))$ is differentiable at $x = a$, what can we conclude about the continuity of $h(x)$ at $x = a$?

If $h(x) = f(g(x))$, and you need to find $h'(3)$ given that $f'(g(3)) = 4$ and $g'(3) = 5$, what is the value of $h'(3)$ using the Chain Rule?

For a nested function defined as $F(x) = e^{[p(q(r(s(t(x)))))]}$ where derivatives at specific points for functions p, q, r, s, t are known, which method provides a more efficient means to find $F'(k)$?

If $h(x) = f(g(x))$ and $f'(x) = 3x^2$ while $g(x) = x^3 + 5$, what is the derivative of $h(x)$ at $x=2$?

If $g(t)= \ln(\cos(t))$, what is $g'(\frac{\pi}{4})$?

Find the derivative of the function $y=(2x+1)^2$ at x=0 using the chain rule.

A function $h(z)$ satisfies $h'(z) = ze^{-z}$. Which expression could represent $h(z)$?

If $f(x) = \cos(2x)$ and $g(x) = f(e^x)$, what is the derivative of $g$ with respect to $x$?

What is the derivative of sin(9x - 8)?