Finding derivatives

Maths: Edexcel A Level Pure Maths, topic 7.1.5

Differentiation

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

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

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

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

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

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 $x$

Test yourself on Finding derivatives

Which notation is correct for the derivative of $y$ = $10x^2$ ?
Which notation is correct for the derivative of $y$ = $2x^5$ ?
Find the derivative of f( $x$ ) = $2x$

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