Differentiation and Its Applications
What the exam tests
Differentiation questions test standard derivatives, product, quotient and chain rules, and economic applications: marginal cost and revenue, slope of a tangent, increasing/decreasing behaviour, and maxima/minima. The exam expects correct notation and a clear second-derivative check where required.
Core rules
- (\frac{d}{dx}x^n=nx^{n-1}), (\frac{d}{dx}e^x=e^x), (\frac{d}{dx}\ln x=\frac1x), (\frac{d}{dx}\sin x=\cos x), (\frac{d}{dx}\cos x=-\sin x).
- Product rule: ((uv)'=u'v+uv'). Quotient rule: ((u/v)'=\frac{u'v-uv'}{v^2}). Chain rule: (\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}).
- Stationary points satisfy (f'(x)=0). If (f''(x)>0), the point is a minimum; if (f''(x)<0), a maximum.
- Marginal cost is (C'(q)); marginal revenue is (R'(q)). Profit is maximised where marginal revenue equals marginal cost under the usual conditions.
Worked example
For (y=x^3-3x^2+2), (\frac{dy}{dx}=3x^2-6x=3x(x-2)). Stationary points are (x=0) and (x=2). The second derivative is (6x-6). At (x=0), (y''=-6<0), so there is a local maximum. At (x=2), (y''=6>0), so there is a local minimum.
Traps
The derivative of a constant is zero. Do not omit the inner derivative in the chain rule. A zero first derivative identifies a stationary point, not automatically an extremum. Keep the sign pattern for increasing/decreasing intervals. In economics, marginal means derivative, not average.