1.7.2—Differentiation rules

Use linearity, the power rule d(xⁿ)/dx=nxⁿ⁻¹, product and quotient rules, and the chain rule for composites. Differentiate with respect to the stated variable.

For a product, u′v+uv′; for a quotient, (u′v−uv′)/v²; for f(g(x)), multiply by g′(x). Simplify after differentiating to avoid losing factors.

d[(x²+1)^3]/dx=6x(x²+1)² by the chain rule.

The derivative of uv is not u′v′, and the chain-rule multiplier is required even when the inner expression looks simple.