Cogito
Calculus · Chapter 4 · Lesson 1
The Differentiation Rules
Shortcuts for what you can already do by hand.
10 problems · about 22 minutes · FUN-3.A, FUN-3.B, FUN-3.C
What this lesson teaches
The student applies the power, product, quotient, and chain rules.
- The power rule drops the exponent and reduces it by one.
- Products and quotients have their own rules; they do not distribute.
- The chain rule multiplies by the derivative of the inside.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Differentiability: y = |x − 2|. At which x is the derivative undefined?
Answer 2
Why x = 2.
Review — The Derivative as a Function: f(x) = x². What is f′(10)?
Answer 20
Why 20.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x⁶. Coefficient in f′(x) = ?x⁵?
Answer 6
Why 6x⁵.
What does the chain rule handle?
Answer A function inside another function.
Why Composition.
f(x) = x⁴. Coefficient in f′(x) = ?x³?
Answer 4
Why The exponent drops down.
f(x) = x³, f′(x) = 3x². What is f′(3)?
Answer 27
Why 3 × 9.
Build It Up
The same ideas with more to keep track of.
2 problemsf(x) = 7x. What is f′(x)?
Answer 7
Why Constant slope.
Is the derivative of a product the product of the derivatives?
Answer No.
Why The product rule has two terms.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Power Rule: f(x) = x⁵. What is the coefficient in f′(x) = ?x⁴?
Answer 5
Why 5x⁴.
The Value: f(x) = x³, so f′(x) = 3x². What is f′(2)?
Answer 12
Why 12.