Derivatives - Clavius

Basic Form

If $y = a x^{n}$ then $y^{'} = a . n . x^{n - 1}$

Example 01

$\begin{aligned} y & = 2 x^{7} \\ y^{'} & = 2 . 7 . x^{7 - 1} \\ y^{'} & = 14 . x^{6} \end{aligned}$

Example 02

$\begin{aligned} y & = 4 x \\ y & = 4 . x^{1} \\ y^{'} & = 4 . 1 . x^{1 - 1} \\ y^{'} & = 4 . x^{0} \\ y^{'} & = 4 \end{aligned}$

Example 03

$\begin{aligned} y & = 5 \\ y & = 5 . x^{0} \\ y^{'} & = 5 . 0 . x^{0 - 1} \\ y^{'} & = 0 \end{aligned}$

A function can be differentiated several times. Second derivative is differentiation of the first derivative.

Example 04

$\begin{aligned} y & = 5 x^{3} \\ y^{'} & = 5 . 3 . x^{3 - 1} (first derivative) \\ y^{'} & = 15 . x^{2} \\ y^{″} & = 15 . 2 . x^{2 - 1} (second derivative) \\ y^{″} & = 30 . x \end{aligned}$

Notations of Derivative

First derivative	$y^{'}$	$\frac{d y}{d x}$
Second derivative	$y^{″}$	$\frac{d^{2} y}{d x^{2}}$

Chain Rule

$\frac{d y}{d x} = \frac{d y}{d u} . \frac{d u}{d x}$

Chain rule can be used to differentiate some functions with different parameter.

Example 05

Given that $y (u) = u^{2} + 5 u$ and $u (x) = 3 x$ . Determine $\frac{d y}{d x}$

$\begin{aligned} y (u) & = u^{2} + 5 u \\ \frac{d y}{d u} & = 2 u + 5 \end{aligned}$

$\begin{aligned} u (x) & = 3 x \\ \frac{d u}{d x} & = 3 \end{aligned}$

$\begin{aligned} \frac{d y}{d x} & = \frac{d y}{d u} . \frac{d u}{d x} \\ \frac{d y}{d x} & = (2 u + 5) . 3 \\ \frac{d y}{d x} & = 6 u + 15 \\ \frac{d y}{d x} & = 6 (3 x) + 15 \\ \frac{d y}{d x} & = 18 x + 15 \end{aligned}$

Another use of chain rule is to differentiate some complex functions.

Example 06

Differentiate $y = (3 x + 2)^{5}$

Method 1

Let $u = 3 x + 2$ , then $y = u^{5}$

$y = u^{5} \to \frac{d y}{d u} = 5 . u^{4}$

$u = 3 x + 2 \to \frac{d u}{d x} = 3$

$\begin{aligned} \frac{d y}{d x} & = \frac{d y}{d u} . \frac{d u}{d x} \\ \frac{d y}{d x} & = 5 . u^{4} . 3 \\ \frac{d y}{d x} & = 15 . u^{4} \\ \frac{d y}{d x} & = 15 . (3 x + 2)^{4} \end{aligned}$

Method 2

$\begin{aligned} y & = (3 x + 2)^{5} \\ \frac{d y}{d x} & = 5 . 3 . (3 x + 2)^{4} \\ \frac{d y}{d x} & = 15 . (3 x + 2)^{4} \end{aligned}$

Exercise

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Basic Form

Chain Rule

Exercise

Differentiation