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Derivatives and differentiation rules

The derivative f′(x) shows how fast a function is changing, which is the gradient of the graph at a point. We differentiate polynomials term by term: xⁿ becomes n · xⁿ⁻¹ and a constant becomes 0. For example, 3x² − 4x + 1 has the derivative 6x − 4.

Maths 3 · Updated

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What does the derivative mean?

The derivative at a point is the gradient of the tangent: how steep the graph is right there. It is also a rate of change, for example speed in metres per second.

A gradient needs two points. We take one more point a distance h away and let h go towards 0.

f′(a) = lim (h → 0) (f(a + h) − f(a)) / h

Differentiation rules for polynomials

We want to differentiate a polynomial one term at a time.

f(x)f′(x)Why
70A constant never changes. The graph is horizontal, gradient 0.
3x3The graph is a straight line with gradient 3.
x²2xThe exponent 2 comes down in front, and the exponent drops to 1.
x³3x²The exponent 3 comes down, and the exponent drops to 2.
4x³12x²The number in front stays: 4 · 3x² = 12x².
xⁿn · xⁿ⁻¹The general rule.

Multiply by the exponent and reduce the exponent by one. A number in front stays where it is. Each term is differentiated on its own.

f′(a) and the tangent

Differentiate first. Then put the number into f′, not into f.

f(x) = 3x² − 4x + 1 gives f′(x) = 6x − 4 and f′(2) = 6 · 2 − 4 = 8

  • f′(a) > 0: the graph is rising at the point.
  • f′(a) < 0: the graph is falling at the point.
  • f′(a) = 0: the tangent is horizontal.
  • The tangent is y = kx + m with k = f′(a), through the point (a, f(a)).

Example

Example 1: differentiate f(x) = 3x² − 4x + 1

We want to find f′(x) by differentiating one term at a time.

  1. The exponent 2 comes down and is multiplied by 3. The exponent drops to 1.

    (3x²)′ = 6x

  2. −4x is a straight line with gradient −4.

    (−4x)′ = −4

  3. A constant never changes, so its derivative is 0.

    (1)′ = 0

  4. We add the derivatives of the terms together.

    f′(x) = 6x − 4

f′(1) = 2. With two nearby points: f(1) = 0 and f(1.01) = 0.0203. The gradient 0.0203 / 0.01 = 2.03 is close to 2.

Example 2: f(x) = x³ − 6x² + 5, work out f′(1) and f′(5)

We want to know the gradient of the graph where x = 1 and x = 5. First we find the derivative.

  1. x³ becomes 3x², −6x² becomes −12x and the constant 5 becomes 0.

    f′(x) = 3x² − 12x

  2. We put x = 1 into the derivative.

    f′(1) = 3 − 12 = −9

  3. We put in x = 5: 3 · 25 = 75 and 12 · 5 = 60.

    f′(5) = 75 − 60 = 15

  4. Negative derivative: the graph is falling. Positive derivative: the graph is rising.

    f′(1) = −9, f′(5) = 15

f(1) = 0 and f(1.001) ≈ −0.009003. The gradient −0.009003 / 0.001 ≈ −9.003 is close to −9.

Example 3: the tangent to f(x) = x² + 2x where x = 1

We want to find the tangent y = kx + m. We need the point and the gradient.

  1. The point: f(1) = 1² + 2 · 1 = 3. The tangent passes through (1, 3).

    f(1) = 3

  2. We differentiate, f′(x) = 2x + 2, and put in x = 1.

    k = f′(1) = 2 · 1 + 2 = 4

  3. We put (1, 3) and k = 4 into y = kx + m.

    3 = 4 · 1 + m

  4. We subtract 4 from both sides: m = 3 − 4 = −1.

    y = 4x − 1

4 · 1 − 1 = 3, so (1, 3) is on the line. x² + 2x = 4x − 1 gives (x − 1)² = 0: only one point in common.

Pitfalls

  • Leaving the constant in, so that the derivative of 3x² + 5 becomes 6x + 5.

    A constant does not change, so its derivative is 0. The answer is 6x.

  • Putting the number into f instead of into f′ when the question asks for the gradient.

    f(a) is where the point is, f′(a) is how steep the graph is there. Differentiate first and then put the number in.

  • Forgetting to reduce the exponent, so that x³ becomes 3x³.

    The exponent comes down and drops by one: x³ becomes 3x², and 4x³ becomes 4 · 3x² = 12x².

Practise on your own

0 of 4 correct

  1. Differentiate f(x) = 5x² + 3x.

  2. Differentiate f(x) = 2x³ − 7x + 4.

  3. f(x) = x² − 6x. Work out f′(5).

  4. Find the equation of the tangent to f(x) = x³ − 2x at the point where x = 2.

Common questions

What is the difference between f(x) and f′(x)?

f(x) is how high the graph is. f′(x) is how steep it is at the same place.

What does it mean when the derivative is zero?

The tangent is horizontal. There the graph can have a maximum, a minimum or a terrace point (a stationary point of inflection). That is why we look for f′(x) = 0 when we search for the largest and smallest values.

What unit does the derivative have?

The unit of y divided by the unit of x. If y is in metres and x is in seconds, we get metres per second, which is a speed.

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