Chain Rule of Differentiation

In differential calculus, the chain rule is a formula for determining the derivative of the combined two or more functions. the chain rule could be used in Leibniz's notation in the following way.
If $y=g(u)$ where $u=f(x)$ then $\displaystyle \dfrac{dy}{dx}=\dfrac{dy}{du} \times \dfrac{du}{dx}$.
Generally, the chain rule is described as following in its simplest understanding.
If $y=\big[f(x)\big]^n$ then $\displaystyle \dfrac{dy}{dx}=n\big[f(x)\big]^{n-1}\times f'(x)$.

Example 1

Find $\displaystyle \dfrac{dy}{dx}$ of $y=(x^2-4x)^5$.

Example 2

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\dfrac{1}{(x^3-3)^6}$.

Example 3

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\sqrt{3x-1}$.

Example 4

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\sqrt[3]{x^2-x}$.

Example 5

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\dfrac{1}{\sqrt{x^3-2x}}$.

Extension Examples

These Extension Examples require to have some prerequisite skills including;
\( \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \dfrac{d}{dx}e^x &= e^x \\ \dfrac{d}{dx}\log_e{x} &= \dfrac{1}{x} \\ \end{align} \)

Example 6

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\sin^3{x}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$.

Example 7

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\cos^3{x}$, known that $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

Example 8

Find $\displaystyle \dfrac{dy}{dx}$ of $y=e^{\sin{x}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}e^x = e^x$.

Example 9

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\sin{e^{2x}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}e^x = e^x$.

Example 10

Find $\displaystyle \dfrac{dy}{dx}$ of $y=\log_e{\sin{e^{x^2}}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\log_e{x} = \dfrac{1}{x}$.