Differentiation

If $y=|\sin x|^{|x|}$, then what is the value of $\frac{d y}{d x}$ at $x=\frac{\pi}{6}$ ?

If $y=\frac{x \sqrt{x^2-16}}{2}-8 \ln \left|x+\sqrt{x^2-16}\right|$, then what is $\frac{d y}{d x}$ equal to?

Paragraph: Consider the equation $x^y=e^{x-y}$ Question: What is $\frac{d y}{d x}$ at $x=1$ equal to ?

Passage: Let $\left(e^y\right)^x-y=0$, where $y$ is a function of $x$ whose domain is $(0,10]$. Question: What is $\dfrac{dy}{dx}$ equal to ?

For the following three (03) items: Let $y = f(x) = \dfrac{x\sin^{-1} x}{\sqrt{1 - x^2}} + \ln\sqrt{1 - x^2}$. If $x = \sin\theta$, then what is $\dfrac{dy}{dx}$ equal to?

If $u=e^{a x} \sin b x$ and $v=e^{a x} \cos b x$, then what is $u \frac{d u}{d x}+v \frac{d v}{d x}$ equal to?

Paragraph: Let $f(x)=x^2+2 x-5$ and $g(x)=5 x+30$ Question: If $h(x)=5 f(x)-x g(x)$, then what is the derivative of $h(x)$ ?

If $f(x)$ is an even function, where $f(x) \neq 0$, then which one of the following is correct?

What is the derivative of $e^x$ with respect to $x^e$ ?

Consider the following for the two (02) items that follow: Let $(x+y)^{p+q} = x^p y^q$, where $p$, $q$ are positive integers. The derivative of $y$ with respect to $x$

Paragraph: Let $f:(-1,1) \rightarrow \mathrm{R}$ be a differentiable function with $f(0)=-1$ and $f^{\prime}(0)=1$. Let $h(x)=f(2 f(x)+2)$ and $g(x)=(h(x))^2$. Question: What is $g^{\prime}(0)$ equal to?

What is the derivative of $2^{(\sin x)^2}$ with respect to $\sin x$ ?