Scottish Higher Maths2 min read

Basic differentiation

Understand the derivative as a gradient function, apply the power rule to polynomials, and evaluate a derivative at a point.

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What differentiation does

A curve can have a different gradient at every point. Differentiation turns the original function into a gradient function: a rule that gives the gradient for any chosen value of $x$ .

If $y=f(x)$ , the derivative may be written as $f'(x)$ or $\frac{dy}{dx}$ . These notations both refer to the gradient function. A value such as $f'(2)$ means the gradient when $x=2$ .

The power rule

Worked example

Differentiate a single power

Differentiate $f(x)=6x^4$ .

Step 1
Bring down the power
Multiply the coefficient by the power: $6\times4=24$ .
Step 2
Reduce the power
Reduce the power from 4 to 3, giving $f'(x)=24x^3$ .

Final answer

$f'(x)=24x^3$

The coefficient and power are multiplied before the power is reduced by 1.

Worked example

Differentiate a polynomial

Differentiate $y=4x^3-5x^2+7$

Step 1
Differentiate the first term
$4x^3$ becomes $12x^2$ .
Step 2
Differentiate the second term
$-5x^2$ becomes $-10x$ .
Step 3
Remove the constant
The derivative of 7 is 0.

Final answer

$\frac{dy}{dx}=12x^2-10x$

Each term is differentiated independently, and the subtraction sign is retained.

Do not leave the constant 7 in the derivative.

Gradient at a point

To find the gradient at a point, differentiate first and then substitute the given $x$ -value into the derivative. Substituting into the original function finds a coordinate, not a gradient.

Worked example

Find a gradient at a point

For $f(x)=x^3+2x$ , find $f'(2)$ .

Step 1
Find the gradient function
Differentiate term by term: $f'(x)=3x^2+2$ .
Step 2
Evaluate at the point
Substitute $x=2$ : $f'(2)=3(2)^2+2=14$ .

Final answer

$f'(2)=14$

The value 14 is the gradient of the curve when $x=2$ .

Self-checkQuick self-check

Differentiate $g(x)=5x^3-4x+9$ , then find $g'(2)$ .

Answer

$g'(x)=15x^2-4,\qquad g'(2)=56$

Differentiate each term, remove the constant, and only then substitute $x=2$ .

What differentiation does

The power rule

Differentiate a single power

Bring down the power

Reduce the power

Differentiate a polynomial

Differentiate the first term

Differentiate the second term

Remove the constant

Gradient at a point

Find a gradient at a point

Find the gradient function

Evaluate at the point