A tangent slope at one point is a number. Leaving the point as a variable turns it into a function that reports the slope everywhere.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The definition
f′(x) is the limit of [f(x + h) − f(x)] ÷ h as h approaches zero.
How to use it
Expand the top, cancel the terms that vanish, divide by h, then let h approach zero.
Notation
f′(x), dy/dx and y′ all mean the same thing. The dy/dx form is a reminder that it is a ratio of changes.
What it reports
Positive means the function is rising, negative means falling, and zero means momentarily flat.
A pattern emerges
The definition applied to x² gives 2x, and to x³ gives 3x². The exponent drops in front and falls by one, which becomes the Power Rule.
One formula for the slope everywhere
Rather than computing the derivative at each point separately, keep the point as a variable. The result is a new function f′ giving the slope at every input, which is a far more powerful object.
Reading f′ off the graph of f
Where f rises, f′ is positive; where f falls, f′ is negative; where f has a horizontal tangent, f′ is zero. Sketching f′ from f is an excellent test of understanding and needs no algebra.
And f from f′
Given f′, you know where f increases and decreases and where it turns, but not its height — that requires one known value. This missing constant is the same one that appears in antidifferentiation.
Derivatives of derivatives
Differentiating f′ gives f″, the rate at which the slope changes. Position, velocity and acceleration are a function and its first two derivatives, which is the clearest illustration of what higher derivatives mean.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1)
- Slope of the tangent2
Step 3: Watch an Example
One step at a time.
Watch Elena Differentiate x² From the Definition
Elena must find f′(x) for f(x) = x² using only the limit definition.
- Step 1
The top is (x + h)² − x², which expands to x² + 2xh + h² − x².
Step 4: Your Turn
Practice makes it stick.
The Slope
Problem 1 of 2
f(x) = x², so f′(x) = 2x. What is f′(7)?
The Line
Problem 2 of 2
f(x) = 5x + 2. What is f′(x)?
From the Definition
1 of 8
f(x) = x². What is f′(3)?
2 of 8
f(x) = x³, so f′(x) = 3x². What is f′(2)?
3 of 8
f(x) = 9. What is f′(x)?
4 of 8
f(x) = −4x. What is f′(x)?
5 of 8
f(x) = x². What is f′(−5)?
6 of 8
f′(x) is negative at x = 2. Is the function rising or falling there? 1 rising, 2 falling.
7 of 8
Put the limit definition process in order.
- 1Expand and cancel the terms without h.
- 2Divide every remaining term by h.
- 3Let h approach zero.
- 4Write f(x + h) − f(x).
8 of 8
f(x) = x³. What is f′(3)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x². What is f′(10)?
Question 2 of 2
What does the derivative of a function report?
What You Learned
- The derivative is the limit of [f(x + h) − f(x)] ÷ h.
- Leaving x as a variable gives a function reporting slope everywhere.
- Its sign says whether the original function is rising or falling.