Methods of Differentiation — NDA Mathematics PYQs

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 ${\left(\frac{dy}{dx}\right)}^2$ 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 $(x^2+4)\frac{dy}{dx}\left[(x^2+4)\frac{d^{2}y}{d{x}^2}-16y\right]$ equal to?

Consider the following for the two (02) items that follow :
Let $(x+y{)}^{p+q}=x^p\cdot y^q$, where $p,q$ are positive integers.

The derivative of $y$ with respect to $x$

Consider the following for the two (02) items that follow :
Let $(x+y{)}^{p+q}=x^p\cdot y^q$, where $p,q$ are positive integers.

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

Consider the following for the two (02) items that follow :
Let $y={\sin }^{-1}\left(x-\frac{4x^3}{27}\right)$.

What is $\frac{dy}{dx}$ equal to?

Consider the following for the three (03) items that follow :
Let $y=f(x)=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}+\ln \sqrt{1-x^2}$.

What is $\frac{d^{2}y}{d{x}^2}$ at $x=0$ equal to?

If $x=\sin \theta$, then what is $\frac{dy}{dx}$ equal to?

Consider the following for the two (02) items that follow :
Consider the function $f(x)=\{\begin{array}{ll}4(5^x),& x<0\\ 8k+x,& x\ge 0\end{array}$

What is $f^{\prime }(-1)$ equal to?

Consider the following for the two (02) items that follow :
Let $u=\int e^x\cos xdx$ and $v=\int e^x\sin xdx$.

What is $u+v$ equal to?

Consider the following for the two (02) items that follow :
Let $u=\int e^x\cos xdx$ and $v=\int e^x\sin xdx$.

Consider the following :
I. $\frac{du}{dx}=-v$
II. $\frac{dv}{dx}=-u$
Which of the above is/are correct?