Differentiation — Finding the Gradient

Differentiation is about finding how fast something is changing. It's one of the two pillars of calculus (the other being integration). Don't panic — the actual mechanics are surprisingly straightforward once you see the pattern.

What IS Differentiation?

Imagine you're driving. Your speedometer tells you how fast you're going right now. That speed is the rate of change of your distance — how quickly the distance is increasing as time passes.

Differentiation does exactly this, but for any equation. If you have a curve y = x², differentiation tells you the gradient (steepness) of that curve at any point.

Key Concept

The derivative of a function gives you a new function that tells you the gradient at any point on the original curve. We write it as dy/dx or f'(x).

The Power Rule

This is the bread and butter of differentiation. For any term of the form ax², you:

Multiply the coefficient by the power
Reduce the power by 1

If y = axⁿ, then dy/dx = nax^n-1

Worked Example 1: Basic Power Rule

Differentiate y = x³

The coefficient is 1, the power is 3
Multiply: 1 × 3 = 3
Reduce power: 3 - 1 = 2
dy/dx = 3x²

Worked Example 2: Multiple Terms

Differentiate y = 5x³ + 2x² - 7x + 4

Differentiate each term separately:

5x³ → 3 × 5 = 15, power 3-1 = 2 → 15x²
2x² → 2 × 2 = 4, power 2-1 = 1 → 4x
-7x → 1 × -7 = -7, power 1-1 = 0 → -7 (since x° = 1)
4 → 0 (constants disappear — they have no x, so there's no change)

dy/dx = 15x² + 4x - 7

Worked Example 3: Negative and Fractional Powers

Differentiate y = 4x² + 3/x

First, rewrite 3/x as 3x^-1:

4x² → 8x
3x^-1 → -1 × 3 = -3, power -1-1 = -2 → -3x^-2 or -3/x²

dy/dx = 8x - 3/x²

Finding the Gradient at a Point

Once you have the derivative, finding the gradient at a specific point is easy — just substitute in the x value.

Worked Example 4: Gradient at a Point

Find the gradient of y = x³ - 2x at the point where x = 2.

Differentiate: dy/dx = 3x² - 2
Substitute x = 2: 3(2)² - 2 = 3(4) - 2 = 12 - 2 = 10

The gradient at x = 2 is 10. The curve is going steeply upwards at that point.

Stationary Points (Turning Points)

A stationary point is where the gradient is zero — the curve is flat for an instant. Think of a ball thrown in the air — at the very top, it's momentarily stationary before coming back down.

To find stationary points: set dy/dx = 0 and solve for x.

Worked Example 5: Finding Stationary Points

Find the stationary points of y = x³ - 3x + 2.

Differentiate: dy/dx = 3x² - 3
Set dy/dx = 0: 3x² - 3 = 0
Solve: 3x² = 3, so x² = 1, so x = 1 or x = -1
Find y values: when x = 1, y = 1 - 3 + 2 = 0. When x = -1, y = -1 + 3 + 2 = 4

Stationary points: (1, 0) and (-1, 4)

Maximum or Minimum?

To find out whether a stationary point is a maximum (hill) or minimum (valley), differentiate again to get d²y/dx²:

If d²y/dx² > 0 at that point → Minimum (the curve is concave up, like a smile)
If d²y/dx² < 0 at that point → Maximum (the curve is concave down, like a frown)

Real World: Speed from Distance

Differentiation isn't just abstract maths. The most common real-world example is motion:

If s(t) gives your distance at time t, then s'(t) (the derivative) gives your speed
Differentiate speed and you get acceleration

Worked Example 6: Speed from Distance

A car's distance (in metres) after t seconds is given by s = 2t³ + 5t. Find the speed at t = 3 seconds.

Differentiate: ds/dt = 6t² + 5
Substitute t = 3: 6(9) + 5 = 54 + 5 = 59 m/s

Common Mistakes

Forgetting to eliminate constants — the derivative of any number on its own (like + 7) is always 0.
Not rewriting before differentiating — terms like 3/x or √x need rewriting as 3x^-1 or x^0.5 first.
Confusing dy/dx with y — dy/dx is the gradient function, not the original curve. Don't substitute into y when you should be substituting into dy/dx.

Practice Questions

Test Yourself

Differentiate y = 4x³
Differentiate y = x² - 6x + 9
Find the gradient of y = 2x² + 3x at x = 4
Find the stationary points of y = x³ - 12x
A ball's height is h = 20t - 5t². Find the time when the ball is at its highest.
Differentiate y = 5/x² (hint: rewrite first)

Answers

dy/dx = 12x²
dy/dx = 2x - 6
dy/dx = 4x + 3. At x = 4: 4(4) + 3 = 19
dy/dx = 3x² - 12 = 0 → x = 2 or x = -2. Points: (2, -16) and (-2, 16)
dh/dt = 20 - 10t = 0 → t = 2 seconds
y = 5x^-2 → dy/dx = -10x^-3 or -10/x³