Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — the quotient rule
Differentiate y=x−32x+1.
- 1Set u=2x+1 and v=x−3.Numerator is u, denominator is v.
- 2u′=2 and v′=1.
- 3dxdy=v2u′v−uv′Subtraction, and top-derivative first.
- 4=(x−3)22(x−3)−(2x+1)(1)
- 5=(x−3)22x−6−2x−1=(x−3)2−7Negative everywhere — consistent with a decreasing hyperbola ✓
Example 2 — stationary points and their nature
Find and classify the stationary points of y=x3−3x2+4.
- 1dxdy=3x2−6x
- 2Set it to zero: 3x(x−2)=0, so x=0 or x=2.Factorise rather than using the formula.
- 3dx2d2y=6x−6
- 4At x=0: −6<0, so a maximum. y=4, giving (0,4).Negative means the curve bends downwards — a hill.
- 5At x=2: 6>0, so a minimum. y=8−12+4=0, giving (2,0).
- 6Maximum (0,4), minimum (2,0).Both y-coordinates are needed — the points, not just the x values.
Example 3 — connected rates of change
The radius of a sphere increases at 0.2 cm s−1. Find the rate of increase of its volume when r=5 cm.
- 1Identify what is given and what is wanted: dtdr=0.2, want dtdV.
- 2The chain that links them: dtdV=drdV×dtdr
- 3V=34πr3, so drdV=4πr2Differentiate before substituting the value of r.
- 4At r=5: drdV=100π
- 5dtdV=100π×0.2=20π≈62.8 cm3s−1Units check: cm2 × cm s−1 = cm3s−1 ✓