Calculus / Unit 3: The Derivative
3.4 The Derivative as a Function
Notes
The derivative $f'$ is itself a function: its input is $x$ and its output is the slope of $f$ at $x$.
- Where $f' > 0$, $f$ is increasing. Where $f' < 0$, $f$ is decreasing. Where $f' = 0$, the tangent is horizontal.
- Results from the limit definition: if $f(x) = x^3$, then $f'(x) = 3x^2$. If $\displaystyle f(x) = \frac{1}{x}$, then $\displaystyle f'(x) = -\frac{1}{x^2}$. If $\displaystyle f(x) = \sqrt{x}$, then $\displaystyle f'(x) = \frac{1}{2\sqrt{x}}$.
- Example: for $f(x) = x^3$, $f'(2) = 3(4) = 12$.
Practice
If $f(x) = x^3$ and $f'(x) = 3x^2$, what is $f'(2)$?
$3(2)^2 = 12$.
For $\displaystyle f(x) = \frac{1}{x}$ we have $\displaystyle f'(x) = -\frac{1}{x^2}$. What is $f'(2)$?
$\displaystyle -\frac{1}{2^2} = -\frac{1}{4}$.
Where $f'(x) < 0$, what is true about the graph of $f$?
Negative slope means the function is going down as $x$ increases.
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