Differentiation · Year 13
In Year 13 you learn: differentiating powers of x.
Differentiation finds the gradient of a curve at any point. This is the rate at which one quantity changes as another changes. The derivative dy/dx is the limit of the gradient of a chord as the chord shrinks to a point. For powers of x it follows a simple rule.
Key facts
- First principles: f′(x) is the limit of (f(x + h) − f(x))/h as h → 0.
- Power rule: if y = ax^n then dy/dx = anx^(n − 1). This works for any real n.
- Differentiate term by term. A constant differentiates to 0.
- Rewrite roots and fractions as powers first: √x = x^(1/2) and 1/(x^2) = x^(−2).
- (d^2y)/(dx^2) is the derivative of dy/dx: the rate of change of the gradient.
Worked examples
Example 1
Find dy/dx for y = 4x^3 − 5x^2 + 7x − 2, and the gradient of the curve at x = 2.
Answer: dy/dx = 12x² − 10x + 7; gradient 35
Example 2
Differentiate y = 6√x − 8/x and find dy/dx when x = 4.
Answer: 2
Example 3
Use first principles to differentiate f(x) = x^2 − 3x.
Answer: f′(x) = 2x − 3