Derivatives of special functions

The essentials

Beyond polynomials and the basic trigonometric functions, you need the derivatives of inverse trigonometric, exponential and logarithmic functions, and three special techniques. Implicit differentiation is used when y is not written alone, as in x² + y² = 25: differentiate every term with respect to x, treating y as a function of x, then solve for dy/dx. Parametric differentiation is used when x and y are both given in terms of a parameter t: dy/dx = (dy/dt) ÷ (dx/dt). Logarithmic differentiation is used for a variable power such as x^x, or for long products and quotients: take log of both sides first, then differentiate.

Key points

• d/dx (sin⁻¹x) = 1/√(1 − x²) and d/dx (cos⁻¹x) = −1/√(1 − x²), for −1 < x < 1.
• d/dx (tan⁻¹x) = 1/(1 + x²) and d/dx (cot⁻¹x) = −1/(1 + x²).
• d/dx (eˣ) = eˣ; d/dx (aˣ) = aˣ log a.
• d/dx (log x) = 1/x for x > 0; with the chain rule, d/dx log(f(x)) = f′(x)/f(x).
• For x² + y² = r², implicit differentiation gives dy/dx = −x/y.
• For y = x^x: log y = x log x, so (1/y) dy/dx = 1 + log x, giving dy/dx = x^x(1 + log x).

Exam tips

• In implicit problems, every y-term picks up a factor dy/dx when differentiated.
• In parametric problems, simplify dy/dt and dx/dt before dividing.

Common mistakes

• Treating x^x as a power (x · x^(x−1)) or as an exponential (x^x log x); it is neither, and needs logarithmic differentiation.
• Forgetting the minus sign in the derivatives of cos⁻¹x and cot⁻¹x.