Differentiation

Differentiation Notation & Basics

Notation 1:

Let’s differentiate the function y = x^{\textcolor{blue}{n}}

The derivative is \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{n}x^{\textcolor{blue}{n}-1}

Example:

Differentiate y = 2x^\textcolor{blue}{3}

\dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{3}\times 2x^{\textcolor{blue}{3}-1} = 6x^2

 

Notation 2:

Let’s differentiate the function f(x) = x^{\textcolor{blue}{n}}

Our derivative, f'(x),=\textcolor{blue}{n}x^{\textcolor{blue}{n}-1}

Example:

Differentiate f(x) = 4x^\textcolor{blue}{9}

f'(x)=\textcolor{blue}{9}\times 4x^{\textcolor{blue}{9}-1} = 36x^8

Differentiation of Linear Combinations

We can also use differentiation to find the derivative of functions involving multiple terms.

To do this, we simply differentiate each term individually:

Example:

Differentiate f(x) = 3x^2 + 4x + 17\\

As the 17 has no x value, we cannot differentiate it so we ignore it…

f'(x) = (\textcolor{blue}{2}\times 3x^{\textcolor{blue}{2}-1}) + (\textcolor{blue}{1}\times 4x^{\textcolor{blue}{1}-1}) + 0\\

f'(x) = 6x + 4

Differentiating Negative Roots

We can differentiate functions with negative roots using the same method.

Example:

Differentiate y = 4x^{-2}

y = 4x^{-2}

Using \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{n}x^{\textcolor{blue}{n}-1},

\dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{-2}\times 4x^{\textcolor{blue}{-2}-1}=-8x^{-3}