Differentiation

Description

Differentiation finds the gradient function (derivative) of a curve — it gives the gradient at any point.

Power rule: $$\frac{d}{dx}(x^n) = nx^{n-1}$$

Examples: $$y = x^3 \Rightarrow \frac{dy}{dx} = 3x^2$$ $$y = 4x^2 + 3x - 7 \Rightarrow \frac{dy}{dx} = 8x + 3$$ $$y = 5 \Rightarrow \frac{dy}{dx} = 0 \quad \text{(constant has zero gradient)}$$

Finding the gradient at a point: substitute $x$ into $\frac{dy}{dx}$.

$$y = x^2 - 3x,\quad x = 4: \quad \frac{dy}{dx} = 2(4) - 3 = 5$$

Stationary points (turning points) occur where $\frac{dy}{dx} = 0$. Solve for $x$, then find $y$.

Nature of stationary points: find $\frac{d^2y}{dx^2}$:

• $\frac{d^2y}{dx^2} > 0$: minimum
• $\frac{d^2y}{dx^2} < 0$: maximum

Equation of tangent at $(x_1, y_1)$: use $y - y_1 = m(x - x_1)$ where $m = \frac{dy}{dx}$ at $x_1$.

Common error: forgetting to reduce the power by 1 — $\frac{d}{dx}(x^3) = 3x^2$, not $3x^3$.

Links

Gradients and intercepts Equation of a straight line