Differentiation
What differentiation is
Differentiation finds the gradient function (derivative) of a curve — the rate at which y changes with x. The derivative is written dy/dx or f′(x), and it gives the gradient of the tangent at any point.
The power rule
For y = axⁿ:
dy/dx = anx^(n−1)"Multiply by the power, then reduce the power by one."
- Example:
y = 3x⁴→ dy/dx = 12x³. - Constants differentiate to 0;
y = x→ dy/dx = 1. - Rewrite roots and fractions as powers first:
√x = x^(1/2),1/x = x⁻¹.
Second derivative
Differentiate again to get d²y/dx² (f″(x)) — the rate of change of the gradient.
Stationary points
At a stationary point, the gradient is zero: dy/dx = 0. Solve this to find their x-coordinates. Classify them using the second derivative:
- d²y/dx² > 0 → minimum (gradient increasing).
- d²y/dx² < 0 → maximum (gradient decreasing).
- = 0 → test further (could be a point of inflection).
Tangents and normals
- Tangent at a point: gradient = dy/dx there; use y − y₁ = m(x − x₁).
- Normal is perpendicular to the tangent: its gradient = −1 ÷ (dy/dx).
Derivatives to learn
| Function | Derivative |
|---|---|
| xⁿ | nx^(n−1) |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
Worked example
Find the stationary point of y = x² − 6x + 5 and determine its nature.
- dy/dx = 2x − 6 = 0 → x = 3; y = 9 − 18 + 5 = −4 → point (3, −4).
- d²y/dx² = 2 > 0 → minimum. ✓
Common mistakes
- Forgetting to reduce the power after multiplying.
- Not rewriting roots/fractions as powers before differentiating.
- Misclassifying stationary points (positive 2nd derivative = minimum).
Exam tips
- Rewrite the function in index form before differentiating.
- Set dy/dx = 0 for stationary points; use the second derivative to classify.
- For a normal, use the negative reciprocal of the tangent gradient.
Key facts to remember
- Power rule: d/dx(axⁿ) = anx^(n−1); learn derivatives of eˣ, ln x, sin x, cos x.
- Stationary points where dy/dx = 0; classify with d²y/dx² (>0 min, <0 max).
- Tangent gradient = dy/dx; normal gradient = −1/(dy/dx).