Calculus: Differentiation and Integration
Core concepts
Differentiation measures instantaneous rate of change. Know the limit definition, the power rule for all real indices, and the product, quotient and chain rules. Interpret $f'(x)$ as gradient and $f''(x)$ as concavity; use these to find stationary points and classify them (sign of $f''$ or a sign table). Applications include tangents and normals, increasing/decreasing intervals, and optimisation problems where you build a function from a constraint, differentiate, solve and verify the maximum or minimum.
Integration reverses differentiation. Know antiderivatives of powers (including $x^{-1}\to\ln|x|$), definite integrals and the Fundamental Theorem. Interpret a definite integral as a signed area; split at $x$-intercepts to compute total area. Apply the trapezoidal rule for approximate areas and understand its over- or under-estimate. Kinematics links displacement, velocity and acceleration through differentiation and integration.
Traps
- Forgetting the chain rule's inner derivative, especially with $\sin$, $\cos$ and exponentials.
- Losing the constant of integration in indefinite integrals.
- Treating a signed integral as total area when the curve crosses the axis.
- Assuming a stationary point is a maximum without checking.
- Confusing average rate of change with instantaneous rate.
Worked example
Differentiate $y=(3x^2+1)^4$. By the chain rule, $y'=4(3x^2+1)^3\cdot 6x=24x(3x^2+1)^3$. Setting $y'=0$ gives $x=0$, the only stationary point.
Rule of thumb
For optimisation, write the quantity to be optimised in terms of one variable, differentiate, solve $f'=0$, then justify with $f''$ or endpoint values.