Defining the Derivative of a Function and Using Derivative Notation
Introduced| English | Meaning |
|---|---|
| derivative/dɪˈrɪvətɪv/ | The instantaneous rate of change of a function with respect to its input. |
The derivative: instantaneous rate, packaged
- The instantaneous rate of change is so useful it gets its own name: the derivative 导数.
- It turns "slope of the tangent at each $x$" into a brand-new function, $f'(x)$.
- Feed it an $x$, and it returns the slope of $f$ right there.
- This lesson makes the definition precise and shows the notations you'll use everywhere.
The derivative is the tangent slope
y = x²
Drag the point along the curve — the number the tangent slope shows is exactly $f'(x)$ at that input.
The limit definition
- The derivative of $f$ is
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$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
- Inside is a difference quotient — the secant slope over an interval of width $h$.
- Taking $h\to0$ shrinks the interval to nothing, turning the secant slope into the tangent slope.
Secant slopes approach the tangent slope: the derivative is the limit of average rates
Secant slopes approach the tangent slope as h shrinks to 0
Which expression is the limit definition of $f'(x)$?
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The derivative is the limit of the difference quotient as $h\to0$.
The "at a point" version
- To find the slope at one specific input $a$, use the alternate form:
-
$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
- Here $x$ slides toward $a$ instead of a step $h$ shrinking to $0$ — same idea, different bookkeeping.
- Both definitions are just "secant slope, in the limit," so they always agree.
You can compute $f'(x)$ by setting $h=0$ directly in $\dfrac{f(x+h)-f(x)}{h}$.
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That gives $\tfrac00$; you must take the limit after simplifying.
The alternate form $f'(a)=\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$ is best described as...
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It is the same tangent slope at $a$, expressed with $x\to a$ instead of $h\to0$.
Reading the notations
- The same derivative wears several outfits — learn to recognize all of them:
- Lagrange: $f'(x)$ (read "$f$ prime of $x$").
- Leibniz: $\dfrac{dy}{dx}$ or $\dfrac{d}{dx}[f(x)]$ — emphasizes "rate of $y$ with respect to $x$."
- All mean the instantaneous rate of change, i.e. the slope of the tangent line at $x$.
The Leibniz notation $\dfrac{dy}{dx}$ means the same thing as $f'(x)$: the ____ of the tangent line.
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Both denote the instantaneous rate of change = tangent slope.
Select all notations that denote the derivative of $y=f(x)$.
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The first three are all the derivative; $f(x)^{-1}$ is a reciprocal/inverse, not a derivative.
You cannot just plug $h=0$ into $\frac{f(x+h)-f(x)}{h}$ — that gives $\tfrac00$. The derivative is the limit as $h\to0$, which you evaluate by simplifying the difference quotient first (cancel the $h$), then letting $h\to0$. Skipping the limit is the classic mistake.
From the definition, $f(x)=x^2$ gives $f'(x)=2x$. Evaluate $f'(3)$.
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$f'(3)=2\cdot3=6$.
Use the definition to find $f'(x)$ for $f(x)=x^2$.
- $\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^2-x^2}{h}=\dfrac{x^2+2xh+h^2-x^2}{h}$.
- $=\dfrac{2xh+h^2}{h}=2x+h$ (for $h\neq0$).
- $\displaystyle\lim_{h\to0}(2x+h)=2x$, so $f'(x)=2x$.
The derivative $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ (or the point form $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$) is the limit of a difference quotient — the slope of the tangent line, i.e. the instantaneous rate of change. Write it as $f'(x)$, $\frac{dy}{dx}$, or $\frac{d}{dx}[f(x)]$.