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The product rule — 0606

Differentiate a product and check by expansion.

Notes and key methods

For \(\displaystyle y=uv, y'=u'v+uv'\). Both terms are needed, even when one factor is linear. Combining this with the chain rule handles products such as \(\displaystyle (ax+b)(x+c)^{n}\).

Differentiate the first derivative carefully to find y″; an independent expansion provides a useful algebraic check.

Check the conditions

Three worked examples

Example 1 · The product rule for algebraic functions

Let \(f(x)=\left(x + 3\right)^{4} \left(4 x + 4\right)\), with real x.

  1. Differentiate the given function, showing the rule used.
  2. Find the second derivative.
  3. Find the equation of the tangent at \(x=0\).
  4. Find the leading coefficient of f′(x).
Show the reasoning

(a)

\(4 \left(x + 3\right)^{4} + 4 \left(x + 3\right)^{3} \left(4 x + 4\right)\) follows by differentiating each factor or inner function as required.

Answer: \(f'(x) = 4 \left(x + 3\right)^{4} + 4 \left(x + 3\right)^{3} \left(4 x + 4\right)\)

(b)

Differentiate the complete first derivative and simplify. \(16 \left(x + 3\right)^{2} \left(5 x + 9\right)\)

Answer: \(16 \left(x + 3\right)^{2} \left(5 x + 9\right)\)

(c)

The point has height \(324\) and gradient \(756\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=756 x + 324\).

Answer: \(y=756 x + 324\)

(d)

Differentiate the leading term \(\displaystyle ax^{n+1}\).

\(20\)

Answer: \(20\).

Example 2 · The product rule for algebraic functions

Let \(f(x)=\left(x + 1\right)^{2} \left(5 x + 5\right)\), with real x.

  1. Differentiate the given function, showing the rule used.
  2. Find the second derivative.
  3. Find the equation of the tangent at \(x=0\).
  4. Find the leading coefficient of f′(x).
Show the reasoning

(a)

\(5 \left(x + 1\right)^{2} + \left(2 x + 2\right) \left(5 x + 5\right)\) follows by differentiating each factor or inner function as required.

Answer: \(f'(x) = 5 \left(x + 1\right)^{2} + \left(2 x + 2\right) \left(5 x + 5\right)\)

(b)

Differentiate the complete first derivative and simplify. \(30 \left(x + 1\right)\)

Answer: \(30 \left(x + 1\right)\)

(c)

The point has height \(5\) and gradient \(15\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=15 x + 5\).

Answer: \(y=15 x + 5\)

(d)

Differentiate the leading term \(\displaystyle ax^{n+1}\).

\(15\)

Answer: \(15\).

Example 3 · The product rule for algebraic functions

Let \(f(x)=\left(x + 2\right)^{3} \left(2 x + 6\right)\), with real x.

  1. Differentiate the given function, showing the rule used.
  2. Find the second derivative.
  3. Find the equation of the tangent at \(x=0\).
  4. Find the leading coefficient of f′(x).
Show the reasoning

(a)

\(2 \left(x + 2\right)^{3} + 3 \left(x + 2\right)^{2} \left(2 x + 6\right)\) follows by differentiating each factor or inner function as required.

Answer: \(f'(x) = 2 \left(x + 2\right)^{3} + 3 \left(x + 2\right)^{2} \left(2 x + 6\right)\)

(b)

Differentiate the complete first derivative and simplify. \(12 \left(x + 2\right) \left(2 x + 5\right)\)

Answer: \(12 \left(x + 2\right) \left(2 x + 5\right)\)

(c)

The point has height \(48\) and gradient \(88\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=88 x + 48\).

Answer: \(y=88 x + 48\)

(d)

Differentiate the leading term \(\displaystyle ax^{n+1}\).

\(8\)

Answer: \(8\).

Common mistakes

Practise with answer checking

Original exam-style problems

Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.

Examination practice · 2020–2026

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