Differentiation

If $\cos\,^{-1}\left(\frac{y}{2}\right)=\log\,_{e}\left(\frac{x}{5}\right)^{5},\left|y\right|<2$, then

Let $f\left(x\right)=\left(\sin\,\left(\tan\,^{-1}x\right)+\sin\,\left(\cot\,^{-1}x\right)\right)^{2}-1$,$\left|x\right|>1$. If $\frac{dy}{dx}=\frac{1}{2}\frac{d}{dx}\left(\sin\,^{-1}\left(f\left(x\right)\right)\right)$…

What is the derivative of $\sin ^2 x$ with respect to $\cos ^2 x$ ?

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

Paragraph: Let $f(x)=|\ln x|, x \neq 1$ Question: What is the derivative of $f(x)$ at $x=0.5$ ?

Paragraph: Let $f(x)=|\ln x|, x \neq 1$ Question: What is the derivative of $f \circ f(x)$, where $1 \lt \mathrm{x} \lt 2$ ?

If $\mathrm{y}=\mathrm{e}^{\mathrm{x}^2} \sin 2 \mathrm{x}$, then what is $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=\pi$ equal to?

What is the derivative of the function $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\tan \mathrm{x}}+\ln (\sec \mathrm{x})-\mathrm{e}^{\ln \mathrm{x}} \text { at } \mathrm{x}=\frac{\pi}{4} ?$

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

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

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?