Calculus / Unit 3: The Derivative
3.3 The Limit Definition of the Derivative
Notes
The derivative of $f$ at $x$ is $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.
- Steps: write $f(x + h)$, subtract $f(x)$, simplify so $h$ cancels, then let $h \to 0$.
- Example: $f(x) = x^2 + 3x$. Then $f(x + h) - f(x) = 2xh + h^2 + 3h$. Divide by $h$ to get $2x + h + 3$. Let $h \to 0$: $f'(x) = 2x + 3$.
Practice
What is $f'(x)$ for $f(x) = 5x - 2$ using the limit definition?
$\displaystyle \frac{5(x + h) - 2 - (5x - 2)}{h} = \frac{5h}{h} = 5$.
Which is the limit definition of $f'(x)$?
The difference quotient compares the output at $x + h$ with the output at $x$, divided by $h$.
Use the limit definition to find $f'(x)$ for $f(x) = x^2 + 3x$.
The difference quotient simplifies to $2x + h + 3$, which approaches $2x + 3$.
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