Gradient of a curve
The derivative dy/dx of a function gives the gradient of its tangent at any point, and the rate at which y changes with x. From first principles, the derivative is the limit of the gradient of a chord as the chord shrinks to a point:
Worked exampleFirst principles
Differentiate f(x) = x² from first principles.
- Form the difference quotient.
- Let h → 0.
Answer: d/dx (x²) = 2x.
Rules of differentiation
| Rule | Formula | Example |
|---|---|---|
| Power | d/dx (xⁿ) = nxⁿ⁻¹ | x⁵ → 5x⁴ |
| Constant multiple | d/dx (kf) = kf′ | 3x² → 6x |
| Sum | (f + g)′ = f′ + g′ | x² + 4x → 2x + 4 |
| Chain | dy/dx = dy/du × du/dx | (2x + 1)⁵ → 10(2x + 1)⁴ |
| Product | (uv)′ = u′v + uv′ | x² sin x → 2x sin x + x² cos x |
| Quotient | (u/v)′ = (u′v − uv′)/v² | x/(x + 1) → 1/(x + 1)² |
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
Worked exampleThe chain rule
Differentiate y = (3x² + 1)⁴.
- Let u = 3x² + 1.
- Multiply.
Answer: dy/dx = 24x(3x² + 1)³.
Partial derivatives
For a function of two variables, z = f(x, y), the partial derivative ∂z/∂x is found by differentiating with respect to x while treating y as a constant, and ∂z/∂y by treating x as constant. For z = x²y + y³: ∂z/∂x = 2xy and ∂z/∂y = x² + 3y².
Try itThe gradient of y = axⁿ
Choose a, n and x. The derivative is anxⁿ⁻¹.
Key terms
- Derivative
- The gradient of a curve at a point.
- Chain rule
- Rule for a function of a function.
- Partial derivative
- A derivative with respect to one variable of several.
- Tangent
- A line that touches the curve and has its gradient.
Summary
- The derivative is the limit of the gradient of a chord.
- Use the power, chain, product and quotient rules.
- Learn the derivatives of sin, cos, tan, eˣ and ln x.
- Partial derivatives treat the other variables as constants.