Differentiation
Differentiation (Extended)
- Differentiation finds the gradient of a curve at any point.
- The rule: if $y = ax^n$, then $\dfrac{dy}{dx} = a\,n\,x^{n-1}$. Differentiate a sum term by term.
- Example: $y = x^3 + 2x^2 - 5x$ → $\dfrac{dy}{dx} = 3x^2 + 4x - 5$.
Practice
If y = x³ + 2x² − 5x, then dy/dx = 3x² + 4x − 5. What is its value at x = 1?
3(1)² + 4(1) − 5 = 3 + 4 − 5 = 2.
Practice
Differentiating y = axⁿ gives:
Bring the power down and reduce it by 1: a·n·xⁿ⁻¹.
Stationary points
- A stationary (turning) point is where $\dfrac{dy}{dx} = 0$.
- Example: $y = x^2 - 6x + 5$ → $2x - 6 = 0$ → $x = 3$, $y = -4$, turning point $(3, -4)$.
Practice
The curve y = x² − 6x + 5 has a turning point. At what x value (dy/dx = 0)?
dy/dx = 2x − 6 = 0 gives x = 3.
You've got it
Key idea
- power rule: $y = ax^n \Rightarrow \dfrac{dy}{dx} = a\,n\,x^{n-1}$
- differentiate a sum term by term
- a stationary point is where $\dfrac{dy}{dx} = 0$