Differential Calculus: Differentiability and Methods of Differentiation
This chapter asks you to identify the correct rule and apply it cleanly. Expect standard derivatives, product, quotient and chain rules, implicit and logarithmic differentiation, parametric forms, and a derivative taken from first principles.
Rules to apply
The derivative is the limit of the difference quotient, and from it come d/dx (xⁿ) = nxⁿ⁻¹, d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec²x, d/dx (sec x) = sec x tan x, d/dx (eˣ) = eˣ, and d/dx (log x) = 1/x. The product rule is (uv)′ = u′v + uv′ and the quotient rule is (u/v)′ = (u′v − uv′)/v². The chain rule multiplies derivatives: dy/dx = (dy/du)(du/dx). For implicit functions differentiate both sides with respect to x and solve for dy/dx. Use logarithmic differentiation for variable powers such as xˣ, and for parametric functions use dy/dx = (dy/dt)/(dx/dt).
Worked example
Differentiate y = sin x/(1 + cos x). By the quotient rule, y′ = [cos x(1 + cos x) − sin x(−sin x)]/(1 + cos x)² = [cos x + cos²x + sin²x]/(1 + cos x)² = (1 + cos x)/(1 + cos x)² = 1/(1 + cos x).
Traps
- Omitting the inner derivative in a chain-rule application.
- Reversing the order of the terms in the quotient rule numerator.
- Treating xˣ as x · x^{x−1} instead of using logarithms.
- Forgetting that dy/dx is undefined where the derivative does not exist, such as |x| at 0.