Differentiation of a Function wrt Another Function and Higher Order derivative of a Function - Practice Questions & MCQ

Differentiation of a Function wrt another Function and Higher Order derivative of a Function

Suppose it is required to differentiate a function $u=f(x)$ w.r.t. another function $v=g(x)$.
Here, $f(x)$ and $g(x)$ are different functions, but the variable i.e. $x$ is the same.
To find du/dv or dv/du, we first differentiate both function $f(x)$ and $g(x)$ w.r.t. $x$ separately and then put these values in the following formulae:

$
\frac{d u}{d v}=\frac{d u / d x}{d v / d x} \quad \text { and } \quad \frac{d v}{d u}=\frac{d v / d x}{d u / d x}
$
Higher Order Derivative of a Function

The derivative of a function is itself a function, so we can find the derivative of a derivative. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, a fourth derivative, and so on. 

Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of $y=f(x)$ can be expressed in any of the following forms:

$
\begin{aligned}
& f^{\prime}(x), f^{\prime \prime}(x), f^{\prime \prime \prime}(x), f^{(4)}(x), \ldots, f^{(n)}(x) \\
& y^{\prime}, y^{\prime \prime}(x), y^{\prime \prime \prime}(x), y^{(4)}(x), \ldots, y^{(n)}(x) \\
& \frac{d y}{d x}, \frac{d^2 y}{d x^2}, \frac{d^3 y}{d x^3}, \frac{d^4 y}{d x^4}, \ldots, \frac{d^n y}{d x^n}
\end{aligned}
$
Note:
It is interesting to note that the notation for $\frac{d^2 y}{d x^2}$ may be viewed as an attempt to express $\frac{d}{d x}\left(\frac{d y}{d x}\right)$ more compactly.
Also $\frac{d}{d x}\left(\frac{d}{d x}\left(\frac{d y}{d x}\right)\right)=\frac{d}{d x}\left(\frac{d^2 y}{d x^2}\right)=\frac{d^3 y}{d x^3}$
$\frac{d^2 y}{d x^2} \neq\left(\frac{d y}{d x}\right)^2$
$\frac{d^2 y}{d x^2} \neq\left(\frac{\frac{d^2 y}{d t^2}}{\frac{d^2 x}{d t^2}}\right)$
$\frac{d^2 y}{d x^2} \neq \frac{1}{\frac{d^2 x}{d y^2}}$