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Differentiation rules

Differentiate products of functions using the product rule, composite functions using the chain rule, and find equations of tangents and normals to curves.

Worked examples

Applying the product rule

Straightforward

Problem

Let f(x)=x3(2x+1)f(x) = x^3(2x + 1). Use the product rule to find f′(x)f'(x), then evaluate f′(−1)f'(-1).

Applying the chain rule to a composite function

Moderate

Problem

Let f(x)=(x2−3)4f(x) = (x^2 - 3)^4. Use the chain rule to find f′(x)f'(x), then evaluate f′(2)f'(2).

Finding equations of the tangent and normal

Challenging

Problem

Find the equations of the tangent and normal to y=x2+2xy = x^2 + 2x at x=1x = 1. Verify that the product of their gradients is −1-1.

Practise

Q1·Straightforward
Let f(x)=x2(x+3)f(x) = x^2(x + 3). Use the product rule ddx[uv]=u′v+uv′\dfrac{d}{dx}[uv] = u'v + uv' with u=x2u = x^2 and v=x+3v = x + 3 to find f′(x)f'(x). State the gradient at x=1x = 1.
Q2·Straightforward
Let f(x)=x3(2x−1)f(x) = x^3(2x - 1). Use the product rule to find f′(x)f'(x), then evaluate f′(2)f'(2).
Q3·Straightforward
Let f(x)=(x+1)(x2−2)f(x) = (x + 1)(x^2 - 2). Use the product rule to find f′(x)f'(x), then evaluate f′(−1)f'(-1).
Q4·Straightforward
Let f(x)=x4(x−5)f(x) = x^4(x - 5). Use the product rule to find f′(x)f'(x), then evaluate f′(1)f'(1).
Q5·Moderate
Let f(x)=(x+2)5f(x) = (x + 2)^5. Use the chain rule dydx=dydu⋅dudx\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} with u=x+2u = x + 2 to find f′(x)f'(x). Evaluate f′(−1)f'(-1).
Q6·Moderate
Let f(x)=(3x2+1)4f(x) = (3x^2 + 1)^4. Use the chain rule to find f′(x)f'(x). Evaluate f′(1)f'(1).
Q7·Moderate
Let f(x)=(2x−3)6f(x) = (2x - 3)^6. Use the chain rule to find f′(x)f'(x). Evaluate f′(1)f'(1).
Q8·Moderate
Let f(x)=(x2−4)3f(x) = (x^2 - 4)^3. Use the chain rule to find f′(x)f'(x). Evaluate f′(3)f'(3).
Q9·Challenging
Find the equation of the tangent to y=x3−xy = x^3 - x at the point where x=2x = 2. State the y-intercept of the tangent.
Q10·Challenging
At the point where x=3x = 3 on y=x2y = x^2, find the gradient of the tangent and then the gradient of the normal. State the gradient of the normal as a fraction. Verify that the product of the two gradients equals −1-1.
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Q11·Challenging
Find the equation of the normal to y=x2−3x+1y = x^2 - 3x + 1 at x=2x = 2. State the x-intercept of the normal.
Q12·Challenging
Find the equation of the normal to y=x3y = x^3 at x=1x = 1. State the y-intercept of the normal as a fraction. Verify that the product of the tangent and normal gradients equals −1-1.
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