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:

$\frac{du}{dv}=\frac{du/dx}{dv/dx}\text{ and }\frac{dv}{du}=\frac{dv/dx}{du/dx}$
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:

$\begin{array}{rl}& f^{\prime }(x),f^{\prime \prime }(x),f^{\prime \prime \prime }(x),f^{(4)}(x),\dots ,f^{(n)}(x)\\ & y^{\prime },y^{\prime \prime }(x),y^{\prime \prime \prime }(x),y^{(4)}(x),\dots ,y^{(n)}(x)\\ & \frac{dy}{dx},\frac{d^{2}y}{d{x}^2},\frac{d^{3}y}{d{x}^3},\frac{d^{4}y}{d{x}^4},\dots ,\frac{d^{n}y}{d{x}^n}\end{array}$
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 $\frac{d}{dx}\left(\frac{dy}{dx}\right)$ more compactly.
Also $\frac{d}{dx}\left(\frac{d}{dx}\left(\frac{dy}{dx}\right)\right)=\frac{d}{dx}\left(\frac{d^{2}y}{d{x}^2}\right)=\frac{d^{3}y}{d{x}^3}$
$\frac{d^{2}y}{d{x}^2}\ne {\left(\frac{dy}{dx}\right)}^2$
$\frac{d^{2}y}{d{x}^2}\ne \left(\frac{\frac{d^{2}y}{d{t}^2}}{\frac{d^{2}x}{d{t}^2}}\right)$
$\frac{d^{2}y}{d{x}^2}\ne \frac{1}{\frac{d^{2}x}{d{y}^2}}$