Differentiation

$$\frac{d}{dx}(ax^n) = anx^{n-1} \qquad \frac{d}{dx}(c) = 0 \qquad \frac{d}{dx}(u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}$$

• Gradient of a curve at a point $=\dfrac{dy}{dx}$ at that point. Equation of the tangent: $y - y_1 = m(x - x_1)$ with $m = \dfrac{dy}{dx}$.
• Turning points: $\dfrac{dy}{dx} = 0$. Maximum if $\dfrac{d^2y}{dx^2} < 0$; minimum if $\dfrac{d^2y}{dx^2} > 0$.
• Rate of change of $y$ with respect to $x$ is $\dfrac{dy}{dx}$. If $s$ is displacement: $v = \dfrac{ds}{dt}$, $a = \dfrac{dv}{dt}$.

Integration

$$\int ax^n\,dx = \frac{ax^{n+1}}{n+1} + c \quad (n \ne -1) \qquad \int a\,dx = ax + c$$ $$\int_a^b f(x)\,dx = F(b) - F(a)$$

• Area under a curve between $x = a$ and $x = b$: $\ \displaystyle\int_a^b y\,dx$. Areas below the $x$-axis come out negative - use the positive value.
• Displacement $= \displaystyle\int v\,dt$, velocity $= \displaystyle\int a\,dt$.

Approximating areas

$$\text{Trapezium rule: } A \approx \frac{h}{2}\left[y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})\right]$$ $$\text{Mid-ordinate rule: } A \approx h\,(y_1 + y_2 + \dots + y_n) \text{ using mid-ordinates}$$ where $h$ is the width of each strip.