The derivative

Find derivatives of monomials and polynomials using the power rule and sum rule, and derive functions from first principles using the limit definition.

Worked examples

Using the power rule to find the gradient at a point

Straightforward

Problem

Let

f(x) = 5x^3

. Find

f'(x)

and state the gradient of the curve at

x = 2

Differentiating a polynomial and evaluating the gradient of the tangent

Moderate

Problem

Let

f(x) = 2x^3 - 5x^2 + 4x - 1

. Find

f'(x)

and the gradient of the tangent at

x = 3

Finding the derivative from first principles

Challenging

Problem

Use the definition

f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}

to find the derivative of

f(x) = x^2 + 3x

Practise

Q1·Straightforward

Let

f(x) = x^4

. Use the power rule

\dfrac{d}{dx}(x^n) = nx^{n-1}

to find

f'(x)

, then state the gradient of the curve at

x = 2

Q2·Straightforward

Let

f(x) = 3x^2

. Find the gradient of the curve at

x = -1

Q3·Straightforward

Let

f(x) = x^5

. Find the gradient of the curve at

x = 1

Q4·Straightforward

Let

f(x) = 2x^3

. Find the gradient of the curve at

x = 3

Q5·Moderate

Let

f(x) = x^3 - 4x^2 + 2

. Differentiate using the sum and scalar multiple rules, then find the gradient of the tangent at

x = 3

Q6·Moderate

Let

f(x) = 2x^4 - 3x^2 + 5x

. Find

f'(1)

Q7·Moderate

Let

f(x) = 4x^2 - x^3 + 1

. Find

f'(-1)

Q8·Moderate

Let

f(x) = x^3 + 2x^2 - 5x + 1

. Find

f'(0)

Q9·Challenging

Use first principles —

f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}

— to find the derivative of

f(x) = x^2

. Then evaluate

f'(-2)

Q10·Challenging

Use first principles to find the derivative of

f(x) = 3x^2

. Then find

f'(2)

Q11·Challenging

Use first principles to find the derivative of

f(x) = x^2 + 5x

. Then evaluate

f'(0)

Q12·Challenging

Use first principles to find the derivative of

f(x) = 2x^2 - 3x

. Then evaluate

f'(1)