Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — implicit differentiation
Find dxdy for x2+xy+y2=7.
- 1Differentiate every term with respect to x.Each y differentiated leaves a dxdy behind.
- 2dxd(x2)=2x
- 3dxd(xy)=y+xdxdyA product of x and y needs the product rule.
- 4dxd(y2)=2ydxdyChain rule — this is where the dxdy comes from.
- 52x+y+xdxdy+2ydxdy=0The right side, a constant, gives 0.
- 6dxdy(x+2y)=−(2x+y), so dxdy=−x+2y2x+yCollect and factorise.
Example 2 — parametric differentiation
A curve is given by x=t2, y=t3. Find dxdy.
- 1dxdy=dx/dtdy/dtThe dt terms cancel formally.
- 2dtdx=2t
- 3dtdy=3t2
- 4dxdy=2t3t2Numerator is the y derivative.
- 5=23tCancel one t; the answer is left in terms of the parameter.
Example 3 — a stationary point with an exponential
Find and classify the stationary point of y=xe−x.
- 1Product rule with u=x, v=e−x.v′=−e−x by the chain rule.
- 2dxdy=e−x−xe−x=e−x(1−x)Factorise before solving.
- 3e−x is never zero, so 1−x=0 and x=1.An exponential factor contributes no solutions.
- 4y=1⋅e−1=e1
- 5dx2d2y=e−x(x−2), which at x=1 is −e1<0.
- 6Maximum at (1,e1).Negative second derivative means maximum.