1.7 Differentiation

The derivative f′(x) is lim_{h→0}[f(x+h)−f(x)]/h when the limit exists. It is the gradient of the tangent and measures instantaneous rate of change.

A secant gradient uses two distinct points; the tangent is its limiting value. At a sharp corner or cusp the two-sided derivative may not exist.

For f(x)=x², the difference quotient is 2x+h, so letting h→0 gives f′(x)=2x.

Substituting h=0 into the quotient before simplifying causes division by zero and is not differentiation.

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.

f′(x) gives the gradient or rate at x; solve f′(x)=m for a specified tangent gradient and f′(x)=0 for stationary candidates. Units carry through the rate interpretation.

Translate the derivative back into the problem: distance gives velocity, velocity gives acceleration, and a stationary candidate still needs classification or endpoint comparison for an optimum.

If s(t)=t³−6t²+9t, then v=s′=3t²−12t+9; stationary position occurs where v=0, but the physical time interval decides which roots matter.

A zero derivative is not automatically a maximum or minimum and may be a stationary inflection.

At a stationary point f′(x)=0. If f″(x)>0 it is a local minimum; if f″(x)<0 a local maximum. When f″=0, inspect the sign of f′ or higher derivatives.

Solve f′=0, calculate coordinates, classify each point and check endpoints if the question asks for an absolute maximum or minimum.

For f=x³−3x, f′=3(x²−1), giving x=±1; f″=6x classifies (−1,2) as a maximum and (1,−2) as a minimum.

Local classification does not compare distant points or endpoints, and f″=0 is inconclusive rather than proof of no turning point.