Cogito
AP Calculus AB · Chapter 3 · Lesson 3
Implicit Differentiation
Differentiating an equation you cannot solve.
12 problems · about 22 minutes · FUN-3.D, FUN-3.E
What this lesson teaches
The student differentiates implicitly and finds tangent lines to implicit curves.
- Differentiate both sides with respect to x, treating y as a function of x.
- Every y term picks up a dy/dx factor from the Chain Rule.
- Collect, factor and divide to isolate dy/dx.
Warm Up
Straightforward practice. Get the method working first.
5 problemsFor x² + y² = 25, what is the slope at (4, 3), as a decimal to two places?
Answer -1.33
Why About −1.33.
Why does differentiating y² give 2y · dy/dx?
Answer y is a function of x, so the Chain Rule supplies its derivative.
Why The Chain Rule, with y as the inner function.
Differentiating y² with respect to x gives 2y times what?
Answer dy/dx
Why The Chain Rule applies.
For x² + y² = 25, slope at (0, 5)?
Answer 0
Why −0 ÷ 5.
For x² + y² = 25, slope at (3, −4), as a decimal?
Answer 0.75
Why −3 ÷ −4.
Build It Up
The same ideas with more to keep track of.
3 problemsx² + y² = 169. Is (5, 12) on the curve? 1 yes, 0 no.
Answer 1
Why 25 + 144.
For x² + y² = 169, slope at (5, 12), as a decimal to two places?
Answer -0.42
Why −5 ÷ 12.
What is the derivative of a constant?
Answer 0
Why It never changes.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the implicit differentiation process in order.
Answer 1. Differentiate both sides with respect to x. 2. Attach dy/dx to every term that came from a y. 3. Collect the dy/dx terms on one side. 4. Factor out dy/dx and divide.
Why The collecting happens after differentiating.
For x² + y² = 100, slope at (6, 8), as a decimal?
Answer -0.75
Why −6 ÷ 8.
The Slope: For x² + y² = 25, dy/dx = −x ÷ y. What is the slope at (3, 4), as a decimal?
Answer -0.75
Why −0.75.
The Other Point: Same curve. What is the slope at (4, 3), as a decimal to two places?
Answer -1.33
Why About −1.33.