Differentiation

Consider the formula for gradient: $$ \begin{equation}\begin{aligned} m&=\frac{y_2-y_1}{x_2-x_1}\\ &=\frac{\Delta y}{\Delta x}\\ \end{aligned}\end{equation} $$

This formula caters for an average rate of change from one point, $(x_1,y_1)$ to another, $(x_2,y_2)$. For differentiation, we find the instantaneous rate of change: $$ \begin{equation}\begin{aligned} m=\frac{dy}{dx}\\ \end{aligned}\end{equation} $$

This is referred to as the gradient function because it produces the gradient of the curve when provided with the $x$ value.

$\Delta$ means an average change whereas $d$ represents an infinitesimally small change - a change so small it is virtually zero ($0$).

Power rule

Given the function $$ \begin{equation}\begin{aligned} f(x)=x^n\\ \end{aligned}\end{equation} $$

The derivative is $$ \begin{equation}\begin{aligned} f'(x)=nx^{n-1}\\ \end{aligned}\end{equation} $$

Power rule practice

Find the first derivatives of the following functions:

• $y=x^4$
• $y=5x^7$
• $f(x)=-11x^{-4}$
• $y=\sqrt{x}$
• $y=\frac{1}{x^3}$

The derivatives of sums and differences

When we have the sum/difference of multiple functions, the derivative is the sum/difference of the derivatives of the individual functions: $$ \begin{equation}\begin{aligned} \frac{d}{dx}[f(x)+g(x)+...]=\frac{d}{dx}[f(x)]+\frac{d}{dx}[g(x)]+...\\ \frac{d}{dx}[f(x)-g(x)-...]=\frac{d}{dx}[f(x)]-\frac{d}{dx}[g(x)]-...\\ \end{aligned}\end{equation} $$

For example: $$ \begin{equation}\begin{aligned} \frac{d}{dx}[x^2+4x]&=\frac{d}{dx}[x^2]+\frac{d}{dx}[4x]\\ &=2x+4\\ \frac{d}{dx}[x^3-5x+7]&=\frac{d}{dx}[x^3]-\frac{d}{dx}[5x]+\frac{d}{dx}[7]\\ &=3x^2-5\\ \end{aligned}\end{equation} $$

Practice

Determine the derivatives of the following:

• $y=4x^3-9x+1$
• $y=-x^7+\sqrt{x}$
• $y=11x^9-\frac{16}{x^2}-12x$
• $f(x)=\frac{-13}{x^5}+\frac{1}{x^2}$

Differentiating trigonometric functions

When differentiating trig functions, we simply use the following identities: