Home / A-Level Maths 9709 / Differentiation / Tangents, normals and rates of change
Cambridge International AS & A Level · Topic 1.7
A-Level Maths 9709 / Topic 1.7 / AS & A Level

Gradients at work.

The derivative gives the gradient of the tangent, so it gives the tangent and the normal too. It also tells you where a function rises or falls, and how fast one quantity changes when another does.

Topic 1.7All differentiation lessons →
The key idea

At a point where the gradient is \(m\), the tangent has gradient \(m\) and the normal has gradient \(-\dfrac{1}{m}\). \(f\) is increasing where \(f'(x) > 0\) and decreasing where \(f'(x) < 0\). Connected rates: \(\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt}\).

x y 1234 (2, 1) y = x³ − 2x² + 1 tangent: y = 4x − 7 normal: x + 4y = 6
Fig. 1: the tangent (gradient 4) and the normal (gradient −1/4) to y = x³ − 2x² + 1 at (2, 1), drawn to equal scales so the right angle is true
Section 01

Tangents and normals.

Three steps every time: find the point (both coordinates), find the gradient by substituting into \(\dfrac{dy}{dx}\), then use \(y - y_{1} = m(x - x_{1})\). For the normal, use the perpendicular gradient \(-\dfrac{1}{m}\) (Fig. 1).

Worked example Tangent and normal to a cubic
Find the equations of the tangent and the normal to \(y = x^{3} - 2x^{2} + 1\) at the point where \(x = 2\).
  1. Point: \(y = 8 - 8 + 1 = 1\), so \((2, 1)\).
  2. Gradient: \(\dfrac{dy}{dx} = 3x^{2} - 4x = 12 - 8 = 4\).
  3. Tangent: \(y - 1 = 4(x - 2)\), so \(y = 4x - 7\).
  4. Normal gradient \(= -\dfrac{1}{4}\): \(y - 1 = -\tfrac{1}{4}(x - 2)\).
  5. Multiply by 4: \(4y - 4 = -x + 2\), so \(x + 4y = 6\).
Answer: Tangent \(y = 4x - 7\); normal \(x + 4y = 6\).
Worked example A normal meeting an axis
The normal to \(y = \sqrt{4x + 1}\) at the point where \(x = 2\) meets the \(x\)-axis at \(A\). Find the coordinates of \(A\).
  1. Point: \(y = \sqrt{9} = 3\), so \((2, 3)\).
  2. Chain rule: \(\dfrac{dy}{dx} = \tfrac{1}{2}(4x + 1)^{-\frac{1}{2}} \times 4 = \dfrac{2}{\sqrt{4x + 1}} = \dfrac{2}{3}\) at \(x = 2\).
  3. Normal gradient \(= -\dfrac{3}{2}\): \(y - 3 = -\tfrac{3}{2}(x - 2)\), i.e. \(3x + 2y = 12\).
  4. On the \(x\)-axis \(y = 0\): \(3x = 12\), \(x = 4\).
Answer: \(A = (4, 0)\)
Your turn
Find the equation of the tangent to \(y = \dfrac{1}{x}\) at the point \(\left(2, \tfrac{1}{2}\right)\).
Examiner trap

The normal gradient is \(-\dfrac{1}{m}\): flip AND change sign. With \(m = 4\), \(-4\) and \(\tfrac{1}{4}\) are both wrong. And never use \(\dfrac{dy}{dx}\) as a formula in the line equation: substitute to get a number first.

Section 02

Increasing and decreasing.

A function is increasing where its gradient is positive and decreasing where its gradient is negative. So find \(f'(x)\), factorise it, and solve an inequality.

Worked example Where is a cubic decreasing?
\(f(x) = x^{3} - 3x^{2} - 9x + 4\). Find the set of values of \(x\) for which \(f\) is decreasing.
  1. \(f'(x) = 3x^{2} - 6x - 9 = 3(x^{2} - 2x - 3) = 3(x - 3)(x + 1)\).
  2. Critical values: \(x = -1\) and \(x = 3\).
  3. \(f'(x)\) is a positive quadratic, so it is negative BETWEEN its roots.
Answer: \(-1 < x < 3\)
Worked example Increasing for all x
Show that \(f(x) = x^{3} + 3x^{2} + 5x\) is an increasing function.
  1. \(f'(x) = 3x^{2} + 6x + 5\).
  2. Complete the square: \(3(x^{2} + 2x) + 5 = 3(x + 1)^{2} - 3 + 5 = 3(x + 1)^{2} + 2\).
  3. \(3(x + 1)^{2} \ge 0\), so \(f'(x) \ge 2 > 0\) for all \(x\).
Answer: \(f'(x) > 0\) for all \(x\), so \(f\) is increasing.
Examiner trap

For a positive quadratic \(f'(x)\), the negative part is between the roots, not outside them. Sketch the parabola of \(f'(x)\) before writing the inequality.

Stage 1 · Learn

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Four quick checks. Each correct answer earns XP and records this skill on your profile.

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The tangent to a curve at \(P\) has gradient \(3\). What is the gradient of the normal at \(P\)?

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Find the equation of the tangent to \(y = x^{2}\) at \((3, 9)\).

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\(f'(x) = x^{2} - 4\). For which values of \(x\) is \(f\) decreasing?

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\(\dfrac{dA}{dr} = 10\) and \(\dfrac{dr}{dt} = 0.3\) at some instant. Find \(\dfrac{dA}{dt}\).

Section 03

Connected rates of change.

When two quantities both change with time, link their rates with the chain rule. Write down the rate you know, the rate you want, and a formula connecting the two quantities; then differentiate the formula.

Worked example A spreading circle
The radius of a circular ripple increases at \(0.5\ \text{cm s}^{-1}\). Find the rate of increase of the area when the radius is \(6\) cm.
  1. Know \(\dfrac{dr}{dt} = 0.5\); want \(\dfrac{dA}{dt}\); link \(A = \pi r^{2}\).
  2. \(\dfrac{dA}{dr} = 2\pi r = 12\pi\) when \(r = 6\).
  3. \(\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt} = 12\pi \times 0.5 = 6\pi\).
Answer: \(6\pi \approx 18.8\ \text{cm}^{2}\,\text{s}^{-1}\)
Worked example Inflating a sphere
Air is pumped into a spherical balloon at \(20\ \text{cm}^{3}\,\text{s}^{-1}\). Find the rate at which the radius is increasing when the radius is \(5\) cm.
  1. Know \(\dfrac{dV}{dt} = 20\); want \(\dfrac{dr}{dt}\); link \(V = \tfrac{4}{3}\pi r^{3}\).
  2. \(\dfrac{dV}{dr} = 4\pi r^{2} = 100\pi\) when \(r = 5\).
  3. \(\dfrac{dr}{dt} = \dfrac{dV}{dt} \div \dfrac{dV}{dr} = \dfrac{20}{100\pi} = \dfrac{1}{5\pi}\).
Answer: \(\dfrac{1}{5\pi} \approx 0.0637\ \text{cm s}^{-1}\)
Examiner trap

Differentiate the formula BEFORE you substitute the particular value. Putting \(r = 5\) into \(V = \tfrac{4}{3}\pi r^{3}\) first turns it into a constant, whose derivative is 0.

Stage 2 · Exam

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The normal to the curve \(y = \sqrt{4x + 1}\) at the point where \(x = 2\) meets the \(x\)-axis at \(A\). Find the \(x\)-coordinate of \(A\).

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\(f(x) = x^{3} - 3x^{2} - 9x + 4\). Find the set of values of \(x\) for which \(f\) is decreasing.

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A spherical balloon is inflated at \(20\ \text{cm}^{3}\,\text{s}^{-1}\). Find the rate of increase of the radius, in \(\text{cm s}^{-1}\), when the radius is \(5\) cm.

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The normal to the curve \(y = 2x^{2} - 3x + 4\) at the point \(P(2, 6)\) meets the \(y\)-axis at \(Q\). Find the \(y\)-coordinate of \(Q\).

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Six original Cambridge-style questions
Equations of tangents and normals, where functions increase or decrease, and connected rates of change for a cube and a conical tank.