Revise
Differentiation
Differentiation Revision
Differentiation
Differentiation is a method used to find the variable rate of change or the gradient of a line. When you differentiate, the result is the derivative, representing the rate of change or gradient.
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}
Example 1: Differentiation
Differentiate the following expressions:
a) y=x^5
b) y=3x^2
c) y=x^{-7}
[6 marks]
a) \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{5} \times x^{\textcolor{blue}{5}-1} = 5x^4
b) \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{2} \times 3x^{\textcolor{blue}{2}-1} = 6x
c) \dfrac{\text{d}y}{\text{d}x}=\textcolor{blue}{-7} \times x^{\textcolor{blue}{-7}-1} = -7x^{-8}
Example 2: Linear Combinations
Differentiate the function f(x) = (x+1)(x+2)
[3 marks]
To solve this, we will firstly expand the brackets:
f(x) = x^2 + 2x + x + 2\\ f(x) = x^2 + 3x + 2\\
We can now differentiate:
f'(x) = (\textcolor{blue}{2}\times x^{\textcolor{blue}{2}-1}) + (\textcolor{blue}{1}\times 3x^{\textcolor{blue}{1}-1}) + 0\\f'(x) = 2x + 3
Differentiation Example Questions
Question 1: Write the derivative of y=2x^4 with respect to x.
[2 marks]
\dfrac{\text{d}y}{\text{d}x} = 8x^3
Question 2: Write the derivative of f(x)=x^{-3} with respect to x.
[2 marks]
Question 3: Write the derivative of f(x)=(x+4)(x+6) - x^3 with respect to x.
[3 marks]
Expand brackets:
f(x) = x^2 + 4x + 6x + 24 - x^3\\ f(x) = -x^3 + x^2 + 10x + 24 \\
Differentiate each term separately:
f'(x) = -3x^{2} + 2x + 10
