Calculus: Differentiation and Applications
Calculus: Differentiation and Applications tests derivative techniques and their use in curve analysis, optimisation, rates and approximations. Expect chain, product, quotient, implicit and parametric differentiation.
Core rules
- Chain: d/dx f(g(x))=f'(g(x))g'(x). Product: (uv)'=u'v+uv'. Quotient: (u/v)'=(u'v-uv')/v².
- Implicit: differentiate both sides with respect to x, treating y as y(x), then solve for dy/dx.
- Parametric: dy/dx=(dy/dt)/(dx/dt), provided dx/dt≠0.
- Stationary points: dy/dx=0; classify by sign change of dy/dx or by the second derivative. Concavity uses d²y/dx².
Worked example
y=x²e^{-x}. By the product rule, y'=2xe^{-x}-x²e^{-x}=x(2-x)e^{-x}. Stationary points occur at x=0 and x=2. For x<0, y'<0; for 0<x<2, y'>0; for x>2, y'<0. Hence x=0 gives a minimum and x=2 gives a maximum.
Traps
Differentiating x²e^{-x} without the product rule; forgetting the derivative of -x; classifying by y=0 rather than dy/dx=0; ignoring domain endpoints; misapplying the parametric quotient; assuming a stationary point is automatically a maximum.