Cogito
Calculus · Chapter 3 · Lesson 2
The Derivative as a Function
One formula giving the slope everywhere.
12 problems · about 22 minutes · CHA-2.B, CHA-2.C
What this lesson teaches
The student computes derivatives from the limit definition and interprets the derivative as a function.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsf(x) = x². What is f′(10)?
Answer 20
Why 20.
What does the derivative of a function report?
Answer The slope of the tangent at each input.
Why The tangent slope at every point.
f(x) = x². What is f′(3)?
Answer 6
Why 2x.
f(x) = x³, so f′(x) = 3x². What is f′(2)?
Answer 12
Why 3 × 4.
f(x) = 9. What is f′(x)?
Answer 0
Why A constant never changes.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = −4x. What is f′(x)?
Answer -4
Why The slope of the line.
f(x) = x². What is f′(−5)?
Answer -10
Why 2 × −5.
f′(x) is negative at x = 2. Is the function rising or falling there? 1 rising, 2 falling.
Answer 2
Why A negative slope descends.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the limit definition process in order.
Answer 1. Write f(x + h) − f(x). 2. Expand and cancel the terms without h. 3. Divide every remaining term by h. 4. Let h approach zero.
Why The division by h cannot happen until the cancellation has.
f(x) = x³. What is f′(3)?
Answer 27
Why 3 × 9.
The Slope: f(x) = x², so f′(x) = 2x. What is f′(7)?
Answer 14
Why 14.
The Line: f(x) = 5x + 2. What is f′(x)?
Answer 5
Why 5.