Methods of Differentiation — NDA Mathematics PYQs

Let $f(x)=\left|\begin{array}{ccc}3x^2& \cos x& -\sin x\\ 6& -1& 0\\ q& q^2& q^3\end{array}\right|$ where $q$ is any constant, then what is $\frac{d^2}{d{x}^2}(f(x))$ at $x=0$ equal to ?

What is the derivative of $\frac{x}{|x|}$ with respect to $x$, where $x<0$ ?

If $\sqrt[4]{y}=x+\sqrt{x^2+4}$, then what is $\sqrt{x^2+4}\frac{dy}{dx}$ equal to ?

Directions : For the next two (02) items that follow :
Let $(e^y{)}^x-y=0$, where $y$ is a function of $x$ whose domain is $(0,10]$.

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

Directions : For the next two (02) items that follow :
Let $(e^y{)}^x-y=0$, where $y$ is a function of $x$ whose domain is $(0,10]$.

What is $\frac{dy}{dx}$ equal to, given that $y=y_0$ when $x=1$ ?

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?