M171 Calculus I · Unit 1: Understanding the derivative · on Exam 1
§1.3 The derivative of a function at a point
Study notes for The derivative of a function at a point in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- The derivative of $f$ at $x=a$ is the limit of average rates of change over $[a,a+h]$ as $h\to0$. It is the slope of the tangent line at $(a,f(a))$ and the instantaneous rate of change of $f$ at $a$.
- Equivalent form: $f'(a) = \lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$.
- Graphically, estimate $f'(a)$ by drawing the tangent line at $(a,f(a))$ and reading its rise over run.
- Working the limit definition always follows the same rhythm: write $f(a+h)$, subtract $f(a)$, simplify until every remaining term has a factor of $h$, cancel the $h$, then let $h\to0$.
Key formulas
Limit definition
$$f'(a) = \lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Alternate form
$$f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
Tangent line at a
$$y = f(a) + f'(a)(x-a)$$
Worked example
Use the limit definition to find $f'(1)$ for $f(x) = \sqrt{2x-1}$.
Show the solution
$f'(1) = \lim_{h\to0}\dfrac{\sqrt{2(1+h)-1}-1}{h} = \lim_{h\to0}\dfrac{\sqrt{1+2h}-1}{h}$. Multiply by the conjugate: $\dfrac{(1+2h)-1}{h(\sqrt{1+2h}+1)} = \dfrac{2}{\sqrt{1+2h}+1} \to \dfrac{2}{2} = 1$. So $f'(1)=1$.
Common mistakes
- Expanding $(a+h)^2$ as $a^2+h^2$. It is $a^2+2ah+h^2$.
- Dropping "$\lim_{h\to0}$" from lines where the limit has not been taken yet, or leaving it after $h$ is gone.
- Plugging $h=0$ before cancelling (gives $\frac00$).
Exam tipWrite "$\lim_{h\to0}$" on every line until you cancel the $h$, then evaluate. Graders look for it.
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