AcademyDifferential Calculus
Academy
Derivative From First Principles
Level 1 - Calculus topic page in Differential Calculus.
Principle
The derivative measures instantaneous rate of change. From first principles, it is the limit of average rates of change over shrinking intervals.
Notation
\(f'(x)\)
derivative of f at x
\(h\)
small change in the input
\(\frac{f(x+h)-f(x)}{h}\)
difference quotient
The Core Method
Use the definition and simplify before taking the limit.
First-principles derivative
\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]
The expression often has a factor of \(h\) that cancels after expanding. Do not substitute \(h=0\) before simplifying.
Worked Cases
Question
Use first principles to differentiate \(f(x)=x^2\).
Answer
Compute the difference quotient: \(\frac{(x+h)^2-x^2}{h}=\frac{x^2+2xh+h^2-x^2}{h}\). Simplify the numerator to \(2xh+h^2=h(2x+h)\). Cancel \(h\) to get \(2x+h\). Taking \(h\to0\) gives \(f'(x)=2x\).
Examples
Question
Use first principles to find the derivative of \(f(x)=3x+1\).
Answer
The difference quotient is \(\frac{3(x+h)+1-(3x+1)}{h}=\frac{3h}{h}=3\). The limit as \(h\to0\) is \(3\).