Differentiation
Core ideas
Differentiate polynomials, exponentials, logarithms, trigonometric functions, and composites using chain, product, and quotient rules. Implicit differentiation, parametric differentiation (dy/dx = (dy/dt)/(dx/dt)), and applications - tangents, normals, stationary points, and the second-derivative test - are core.
Traps
- Forgetting the chain rule's inner derivative (e.g. d/dx sin(3x) = 3 cos(3x)).
- The quotient rule sign: (u'v - uv')/v^2, not (uv' - u'v)/v^2.
- In parametric differentiation, divide by dx/dt, do not multiply.
- The second derivative being zero does not guarantee a point of inflection; check sign change.
- Implicit differentiation requires applying the chain rule to y terms (d/dx of y^2 is 2y dy/dx).
Rule of thumb
For composite functions, work outside-in: differentiate the outer function, keep the inner unchanged, then multiply by the derivative of the inner. Write the chain explicitly to avoid dropped factors.
Worked example
Differentiate y = x^2 e^(3x). Use the product rule with u = x^2, v = e^(3x). Then u' = 2x, v' = 3e^(3x). So dy/dx = 2x e^(3x) + x^2 · 3e^(3x) = e^(3x)(2x + 3x^2) = x e^(3x)(2 + 3x). Setting dy/dx = 0 gives x = 0 or x = -2/3; the second derivative test confirms a minimum at x = 0 and a maximum at x = -2/3.