Math Bytes · Calculus · Episode 5 · "Derivatives: definition and basic rules"
The derivative as a limit
Zoom in on a smooth curve and it looks like a straight line. The derivative is the slope of that line, found as the limit of the slope between x and x + h as h shrinks to zero.
Zoom in on a curve
Zoom in on a smooth curve, again and again, and it starts to look like a straight line. That straight line is the tangent line, and its slope is the derivative at that point. The question is how to find that slope with algebra, since you need two points to compute a slope.
Two points, then shrink the gap
Pick two points on the graph of f: one at x and one a little further along, at x + h. The slope of the line through them (a secant line) is rise over run:
slope = [f(x + h) − f(x)] / h
This is called the difference quotient. Now slide the second point toward the first by shrinking h toward zero. The secant line tilts and settles into the tangent line. The derivative is the limit of that slope:
f′(x) = lim h→0 [f(x + h) − f(x)] / h
Worked example: f(x) = x² at x = 3
Since f(3) = 9, the difference quotient is:
| Step | Expression | |---|---| | Start | [(3 + h)² − 9] / h | | Expand (3 + h)² | [9 + 6h + h² − 9] / h | | The 9s cancel | (6h + h²) / h | | Divide by h | 6 + h |
As h goes to zero, 6 + h goes to 6. So f′(3) = 6: the tangent line to y = x² at the point (3, 9) has slope 6. That matches the last lesson, where the average speeds 7, 6.1 and 6.01 closed in on 6. Those were exactly 6 + h for h = 1, 0.1 and 0.01.
Try it yourself
Use the same f(x) = x², but at x = 5. Since f(5) = 25, write the difference quotient [(5 + h)² − 25] / h, expand, cancel and divide by h. What does it head to as h goes to zero? (Answer: (10h + h²) / h = 10 + h, which goes to 10, so f′(5) = 10.)
Good to know
- "Zoom in and it looks like a line" works where the function is differentiable (smooth). A curve with a corner, like the graph of |x| at 0, never straightens out, so it has no derivative there.
- Cancelling h is allowed because h is never exactly zero inside the limit; we only ask what the expression approaches.
- The notation f′(x) and dy/dx mean the same thing for y = f(x).
- Next lesson, the power rule gives these answers in seconds, with no limit work by hand.
Answer to the comment question
f(x) = x². What's f′(4)? f′(4) = 8. Since f(4) = 16: [(4 + h)² − 16] / h = (16 + 8h + h² − 16) / h = (8h + h²) / h = 8 + h, which goes to 8 as h goes to zero.
Test your knowledge
Quick quiz: The derivative as a limit
4 multiple-choice questions. Pick one answer for each, then check how you did.
Sources
- OpenStax, "Calculus Volume 1, 3.1 Defining the Derivative": the limit of the difference quotient; the tangent line as the limit of secant lines
- OpenStax, "Calculus Volume 1, 3.2 The Derivative as a Function": f′(x) = lim h→0 [f(x + h) − f(x)] / h; differentiability
- College Board, "AP Calculus AB and BC Course and Exam Description": Unit 2, defining the derivative with a limit; the difference quotient
- Wolfram MathWorld, "Derivative": limit definition of the derivative
- MIT OpenCourseWare, "18.01 Single Variable Calculus (Fall 2006)": lecture notes on derivatives, the difference quotient and tangent lines
Video transcript
What do you see if you zoom in on a curve forever?
Zoom in on a smooth curve and it starts to look like a straight line. Its slope is the derivative.
Pick two points, x and x + h. Their slope is f(x + h) − f(x), over h.
Then shrink h to zero. f′(x) is the limit of that slope as h approaches zero.
Try f(x) = x² at 3. (3 + h)², minus 9, over h, simplifies to 6 + h.
As h goes to zero, that's 6. So f′(3) = 6.
Your turn. Same f(x) = x². What's f′(4)? Comment your answer below!
That's your byte.