2.4 Differentiation

Differentiate e^{g(x)} as g′(x)e^{g(x)}, ln(g(x)) as g′(x)/g(x), and sin or cos of an inner function using the chain rule.

Identify the outer function first, differentiate it, then multiply by the derivative of the inner expression. Simplify only after all factors are present.

d[ln(1+x²)]/dx=2x/(1+x²), while d[e^{3x}sin x]/dx=e^{3x}(3sin x+cos x).

The derivative of ln g is not 1/ln g, and forgetting g′ changes the scale of the answer.

For y=uv, y′=u′v+uv′. For y=u/v, y′=(u′v−uv′)/v². Both rules follow from the product rule and reciprocal structure.

Label u and v before differentiating, keep the denominator squared in the quotient rule, and use a logarithmic derivative when products of powers are cumbersome.

d[x²e^x]/dx=e^x(x²+2x); d[(sin x)/x]/dx=(x cosx−sinx)/x².

Differentiating numerator and denominator separately is not the quotient rule.

A composite expression may require chain, product, quotient or implicit differentiation. Read its structure before applying a familiar rule.

Rewrite constants and powers clearly, differentiate one layer at a time, and check the result by estimating the sign or scale of the gradient at a simple point.

For y=(x²+1)^4/(x−1), treat the numerator and denominator as factors and apply product/quotient plus chain rules rather than expanding blindly.

There is no single “power rule” shortcut for a quotient or a composite power without the required extra factors.