Exponential, log and logarithmic differentiation

Two derivatives to know by heart:

• d/dx eˣ = eˣ: the exponential function is its own derivative.
• d/dx log x = 1/x (for x > 0). In calculus, log means the natural log, base e.

With the chain rule: d/dx e^(3x) = 3e^(3x), and d/dx log(sin x) = cos x / sin x = cot x.

Log rules turn hard operations into easy ones:

• log(pq) = log p + log q (product → sum);
• log(p/q) = log p – log q (quotient → difference);
• log pⁿ = n log p (power → product).

Logarithmic differentiation uses these rules. Use it when:

• the power itself contains x, as in x^(sin x) or (log x)ˣ;
• there is a long product or quotient of many factors.

Method: take log of both sides, simplify with the rules, differentiate (the left side gives (1/y) dy/dx), then multiply by y.

Example: y = x^(sin x). log y = sin x · log x. Differentiate: (1/y) dy/dx = cos x · log x + sin x · (1/x). So dy/dx = x^(sin x) (cos x log x + sin x / x).

If y is a sum of such terms, like xˣ + (sin x)ˣ, give each term its own letter, u and v, and differentiate them separately. The log of a sum does not split.