Derivatives in applications
Second-order, parametric and implicit derivatives; rates of change.
Revisiting derivative techniques
Second-order derivatives
If y = f(x), then d²y/dx² is the derivative of dy/dx. It measures how fast the slope itself changes and is used in the second-derivative test and in describing acceleration. For y = x³ − 4x: dy/dx = 3x² − 4 and d²y/dx² = 6x, which equals 12 at x = 2.
Parametric differentiation
If x = f(t) and y = g(t), then dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. For the second derivative, differentiate dy/dx with respect to t and divide again by dx/dt: d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt). Example: x = a cos t, y = a sin t gives dy/dx = −cot t and d²y/dx² = cosec²t ÷ (−a sin t) = −cosec³t/a.
Implicit differentiation
When x and y are linked by an equation such as xy = 12, differentiate both sides with respect to x, treating y as a function of x and using the chain rule: y + x(dy/dx) = 0, so dy/dx = −y/x. At (3, 4), dy/dx = −4/3.
Rates of change
If a quantity Q depends on x and x changes with time t, then dQ/dt = (dQ/dx)(dx/dt). Example: the radius of a spherical balloon increases at 2 cm/s. Since V = (4/3)πr³, dV/dt = 4πr² × dr/dt. At r = 5 cm, dV/dt = 4π(25)(2) = 200π cm³/s.
In business, the same idea gives the rate of change of cost, revenue or profit with respect to output or time.
Key points
- d²y/dx² = d/dx(dy/dx).
- Parametric: dy/dx = (dy/dt)/(dx/dt); for d²y/dx² divide by dx/dt again.
- Implicit: differentiate term by term, multiply each derivative of y by dy/dx.
- Related rates: chain rule dQ/dt = (dQ/dx)(dx/dt).
Common mistakes
- Writing d²y/dx² = (d²y/dt²)/(d²x/dt²) for parametric curves; this is wrong.
- Forgetting the dy/dx factor when differentiating y² or xy implicitly.
- Substituting numerical values before differentiating in rate problems.