Cogito
AP Calculus AB · Chapter 3 · Lesson 1
Power, Product, and Quotient Rules
Differentiating without the limit each time.
11 problems · about 24 minutes · AP Calculus AB 2.5, 2.8, 2.9
What this lesson teaches
The student applies the power, product, and quotient rules.
- The power rule brings the exponent down and reduces it by one.
- The product rule is f'g + fg', not the product of derivatives.
- The quotient rule subtracts in a fixed order, over the denominator squared.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x³. f'(3)?
Answer 27
Why 27.
f(x) = x⁵. f'(1)?
Answer 5
Why 5x⁴ at 1.
f(x) = 3x². f'(2)?
Answer 12
Why 6x.
f(x) = x⁻¹. The derivative is −x⁻². What is f'(1)?
Answer -1
Why −1 over 1.
Build It Up
The same ideas with more to keep track of.
3 problemsMatch each rule to its formula.
Answer Power rule → n·xⁿ⁻¹; Product rule → f'g + fg'; Quotient rule → (f'g − fg') / g²
Why Only the quotient rule has a denominator.
f(x) = x² + x³. f'(1)?
Answer 5
Why 2x + 3x² at 1.
Is (fg)' equal to f'g'? 1 for yes, 0 for no.
Answer 0
Why Test with f = g = x.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsf(x) = 10x. f'(4)?
Answer 10
Why A line.
f(x) = x⁷. f'(1)?
Answer 7
Why 7x⁶ at 1.
The Power Rule: f(x) = x⁶. What is the coefficient in the derivative?
Answer 6
Why 6x⁵.
Evaluated: f(x) = x⁴. What is f'(2)?
Answer 32
Why 4 × 8 = 32.