Differentiation

Let $f(x)$ and $g(x)$ be two functions such that $g(x)=x-\frac{1}{x}$ and $f$ o $g(x)=x^3-\frac{1}{x^3}$. What is $f^{\prime}(x)$ equal to?

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. If $p + q = 10$, then what is $\dfrac{dy}{dx}$ equal to?

If $y=\left(x^{\mathrm{x}}\right)^{\mathrm{x}}$ then which one of the following is correct?

Paragraph: Let $f(x)=|\ln x|, x \neq 1$ Question: What is the derivative of $f(x)$ at $x=2$ ?

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, given that $y=y_0$ when $x=1$ ?

What is the derivative of $\operatorname{cosec}\left(x^{\circ}\right)$ ?

If $y=\tan ^{-1}\left(\frac{5-2 \tan \sqrt{x}}{2+5 \tan \sqrt{x}}\right)$, then what is $\frac{d y}{d x}$ equal to?

If $x^y y^x=1$, then what is $\frac{d y}{d x}$ at $(1,1)$ equal to?

Consider the following for the two (02) items that follow: Let $x = \sec\theta - \cos\theta$ and $y = \sec^4\theta - \cos^4\theta$. What is $\left(\dfrac{dy}{dx}\right)^2$ equal to?

If $e^{\theta \varphi}=c+4 \theta \varphi$, where $c$ is an arbitrary constant and $\varphi$ is a function of $\theta$, then what is $\varphi \mathrm{d} \theta$ equal to?

Let $f(x)$ be a polynomial function such that $f \circ f(x)=x^4$. What is $f^{\prime}(1)$ equal to?

Paragraph: Consider the equation $x^y=e^{x-y}$ Question: What is $\frac{d^2 y}{d x^2}$ at $x=1$ equal to?