C1: Differentiation and integration

The derivative of $f(x)$ at a point is the gradient of the tangent to $y=f(x)$ there. It is also an instantaneous rate of change. Standard notations include $f'(x)$, $f''(x)$, $\frac{dy}{dx}$, and $\frac{d^2y}{dx^2}$.

Differentiate sums and differences term by term. The syllabus covers rational powers, sine, cosine, tangent, $e^x$, and $\ln x$, together with constant multiples. Products use the product rule, quotients use the quotient rule, and composite functions use the chain rule.

A differentiable function is increasing where $f'(x)>0$ and decreasing where $f'(x)<0$. Stationary points satisfy $f'(x)=0$. Use the sign change of the first derivative or the second derivative to distinguish a maximum, minimum, or stationary point of inflexion. The second derivative test is inconclusive when $f''(x)=0$.

Use the derivative for gradients, tangents, normals, connected rates, and optimisation. A normal is perpendicular to the tangent. In connected-rate problems, relate the changing quantities before differentiating with respect to time. In optimisation, solve the stationary condition and justify the required extreme.

Integration reverses differentiation and introduces an arbitrary constant for an indefinite integral. The syllabus includes rational powers, sine, cosine, $\sec^2x$, and $e^x$, plus corresponding forms with a linear argument $ax+b$. Differentiate an antiderivative to check the factor introduced by the inner linear function.

A definite integral evaluates an antiderivative at its upper and lower limits. It gives signed area: regions below the horizontal axis contribute negatively. For geometric area bounded by a curve and line or lines, split at intersections and axis crossings, then reverse the sign of any below-axis contribution. Area between two curves is excluded from this syllabus scope.

For displacement $s(t)$, velocity is $v=\frac{ds}{dt}$ and acceleration is $a=\frac{dv}{dt}=\frac{d^2s}{dt^2}$. Integration reverses these relations but requires constants from initial conditions. Interpret velocity sign as direction and distinguish displacement from total distance.