Differentiation and Applications
Core concepts
You must differentiate polynomials, products, quotients, composite functions (chain rule), and trigonometric, exponential and logarithmic functions. Know the derivative as a limit, the interpretation as a gradient, and the applications: tangents and normals, stationary points with the second-derivative or sign test, increasing/decreasing intervals, and rates of change including connected rates.
Traps
- The chain rule is frequently misapplied; identify inner and outer functions explicitly.
- The quotient rule sign is −(u'v − uv')/v², not the reverse.
- For stationary points, classify them; simply stating 'stationary' is not enough.
- In connected rates, use dy/dt = dy/dx · dx/dt and substitute at the correct instant.
- Differentiating sin x or cos x in degree mode gives wrong answers; calculus requires radians.
Worked example
Differentiate y = x² e^(3x). Use the product rule with u = x², v = e^(3x): u' = 2x, v' = 3e^(3x). Then y' = 2x e^(3x) + 3x² e^(3x) = x e^(3x)(2 + 3x). Rule of thumb: when two factors appear, factor out the exponential at the end for a neat answer.