3.4 Differentiation

For a composite y=f(g(x)), y′=f′(g(x))g′(x). Exponential, logarithmic and trigonometric outer functions all use this chain-rule structure.

Label the inner function and its derivative explicitly. For products or quotients, combine the product/quotient rule with the chain rule rather than expanding unnecessarily.

d[sin((2x+1)²)]/dx=4(2x+1)cos((2x+1)²).

The inner derivative cannot be omitted just because the outer derivative looks familiar.

For y=uv use u′v+uv′; for y=u/v use (u′v−uv′)/v². The formulas account for both numerator and denominator variation.

Choose factors that keep algebra manageable, and simplify only after applying the rule. A logarithmic derivative can be shorter for a product of powers.

d[x sin x]/dx=sin x+x cos x; d[(x+1)/(x−1)]/dx=−2/(x−1)².

Differentiating top and bottom independently is invalid for a quotient.

Power, exponential, logarithmic, trigonometric, product, quotient and chain rules are building blocks; a single expression may require several in sequence.

Rewrite constants and nested powers, mark the outermost operation, and check a derivative numerically at one simple point if the algebra is long.

For y=(x²+1)^2e^x, use product plus chain: y′=e^x[(x²+1)^2+4x(x²+1)].

A familiar-looking power does not override a product or composite structure; missing one rule changes the result.