Derivatives from local change

What this concept is

The derivative f′(x) is the limit as h tends to zero of [f(x+h)−f(x)]/h, when that limit exists. It is the local rate of change and tangent slope.

Why it matters here

Performance sensitivity and optimization depend on change in output per unit of input, not just the output itself.

Concrete example

For f(x)=x², the difference quotient is (2xh+h²)/h=2x+h for h≠0; its limit is 2x. At x=3 the tangent line is y=9+6(x−3). For distance in metres versus time in seconds, the derivative has units metres/second.

Common confusion / misconception

Continuity does not imply differentiability: |x| is continuous at zero but its left and right slopes are −1 and 1. A numerical slope at a single step size is not the derivative definition.

How to use it

Form the difference quotient, simplify before taking a limit, and check the domain and one-sided slopes. The power rule d(x^n)/dx=n x^(n−1) applies to integer powers where defined. Differentiation is linear: constants differentiate to zero and scalar multiples/sums differentiate termwise.

Check yourself

Derive the derivative of 3x²−2x+4 from the definition or linearity. What is its slope at x=1?

Mini drill or application

Derive d(1/x)/dx=−1/x² for x≠0 by combining fractions. Explain why x=0 cannot be repaired by this derivative formula. Compare secant slopes at x=2 for h=0.1 and 0.01 with the exact slope of x².

Read this only if stuck

OpenStax selective support. Use the reading guide for the section sequence and practice contract. Record your reasoning before consulting solutions.