Derivatives of Functions

Maths: Edexcel IGCSE Higher, topic 3.5.1

Derivatives

The derivative of a function, f(x), is the function that describes how the gradient changes as x changes. This is the same as saying that the derivative the rate of change of an equation.

Derivatives

The derivative of a function can be used to calculate the gradient of the function at any point.
The derivative of f( $x$ ) is called f'( $x$ ).
f'( $x$ ) can be called "f dash of $x$ " or "f prime".

Example

A function, f( $x$ ), has the derivative, f'( $x$ ) = 2 $x$ + 5
What is the gradient of f( $x$ ) when $x$ = 3?
Gradient at $x$ = 3 is f'(3).
f'(3) = (2 × 3) + 5 = 11
f( $x$ ) has a gradient of 11 when $x$ = 3.

Differentiation

The derivative of a function can be worked out through a process called differentiation.

Differentiation

There are different ways to display the derivative of a function.
If your equation starts with y =, then the derivative of the equation is $\large\frac{dy}{dx}$ .
If your function starts with f( $x$ ) =, then the derivative of the equation is f'( $x$ ).
The exam may also ask you to find the rate of change, the gradient of the tangent or to differentiate a function.

General rule

To differentiate a whole equation/function, each term should be differentiated individually.
The general rule for differentiating a term is:
f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}

Examples

f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}
f( $x$ ) = $x$ ³ → f'( $x$ ) = 3 $x$ ²
f( $x$ ) = 2 $x$ ³ → f'( $x$ ) = 6 $x$ ²
f( $x$ ) = 6 $x$ → f'( $x$ ) = 6 (remember f( $x$ ) = 6 $x$ is the same as 6 $x$ ¹)
f( $x$ ) = 3 $x$ ⁴ → f'( $x$ ) = 12 $x$ ³
f( $x$ ) = 9 → f'( $x$ ) = 0 (remember f( $x$ ) = 9 is the same as 9 $x$ ⁰)

Example 1 with multiple terms

For each term in an expression, use f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}
f( $x$ ) = $x$ ³ + 2 $x$ ² + 7 $x$
f'( $x$ ) = 3 $x$ ² + 4 $x$ + 7

Example 1 with multiple terms

For each term in an expression, use f( $x$ ) = $ax^n$ → f'( $x$ ) = $nax$ ^{$n$ −1}
f( $x$ ) = 3 $x$ ⁴ + 5 $x$ ³ + 4 $x$ ²
f'( $x$ ) = 12 $x$ ³ + 15 $x$ ² + 8 $x$

Derivatives in Kinematics

Another use for the derivative is to analyse motion along a line.

Velocity

The velocity of an object in motion is the rate of change of its displacement.
So the derivative of an object's displacement as a function of time is equal to the velocity, or:
v(t) = s'(t)
Where v(t) is the velocity and s(t) is the position, both as a function of time

Acceleration

Taking the derivative of the velocity gives the acceleration of the object, which is the rate of change of velocity.
a(t) = v'(t) = s''(t)
So the acceleration can also be thought of as the second derivative of the object's displacement with respect to time.

Test yourself on Derivatives of Functions

What are the ways you can say f'(x) out loud?
The derivative of a function, f(x), is f'(x) = 5x + 10. What is the gradient of f(x) when x = −1?
The derivative of a function, f(x), is f'(x) = $\large\frac{1}{2}$ x² − 3x + 2. What is the gradient of f(x) when x = 4?
The derivative of a function, f(x), is f'(x) = 2x − 1. What is the gradient of f(x) when x = 2?
The derivative of a function, f(x), is f'(x) = 9x³ − 10x. What is the gradient of f(x) when x = −1?

Practise this topic for free