M171 Calculus I · Unit 1: Understanding the derivative · on Exam 1
§1.4 The derivative function
Study notes for The derivative function 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
- Letting the point vary gives a new function $f'(x)$, the derivative function: $f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$.
- Reading $f'$ from the graph of $f$: where $f$ rises, $f'>0$; where $f$ falls, $f'<0$; at a peak or valley (smooth), $f'=0$; at a corner or cusp, $f'$ is undefined; steeper $f$ means larger $|f'|$.
- Sketching $f'$: mark the zeros first (where $f$ has horizontal tangents), decide the sign on each interval, then estimate steepness.
- Notation: $f'(x)$, $\dfrac{dy}{dx}$, $\dfrac{d}{dx}[f(x)]$, $y'$.
Key formulas
Derivative function
$$f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$
Worked example
Find $r'(x)$ for $r(x) = 11 - 5x - 3x^2$ using the definition.
Show the solution
$r(x+h) = 11 - 5x - 5h - 3x^2 - 6xh - 3h^2$. Subtract $r(x)$: $-5h - 6xh - 3h^2$. Divide by $h$: $-5 - 6x - 3h$. Let $h\to0$: $r'(x) = -5 - 6x$.
Common mistakes
- Sketching $f'$ as a shifted copy of $f$. The graph of $f'$ records slopes, not heights.
- Forgetting that $f'$ is undefined at sharp corners, e.g. $|x|$ at $x=0$.
Exam tipA helpful sentence to use: "Since $f$ is increasing on $(a,b)$, $f'(x)>0$ there." Connecting words like this are what the rubric calls explaining your thinking.
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