Lesson 4: Implicit Differentiation
Estimated time: 35-45 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Distinguish between explicit and implicit functions
- Apply the technique of implicit differentiation
- Find tangent lines to implicitly defined curves
- Use implicit differentiation to find d²y/dx²
Explicit vs. Implicit Functions
An explicit function is written y = f(x), like y = x² + 3. An implicit relation defines a relationship between x and y without isolating y, such as x² + y² = 25 (a circle).
Many important curves cannot be written as a single explicit function. Implicit differentiation lets us find dy/dx directly from the equation.
The Technique
Implicit Differentiation Procedure
- Differentiate both sides of the equation with respect to x.
- Every time you differentiate a term containing y, apply the Chain Rule: d/dx [f(y)] = f'(y) · dy/dx.
- Collect all terms with dy/dx on one side.
- Factor out dy/dx and solve.
Example 1: Circle
Find dy/dx for x² + y² = 25.
Step 1: Differentiate both sides: 2x + 2y(dy/dx) = 0.
Step 2: Solve for dy/dx: 2y(dy/dx) = −2x, so dy/dx = −x/y.
Interpretation: At the point (3, 4), the slope is −3/4. At (3, −4), the slope is 3/4.
Example 2: Product of x and y
Find dy/dx for xy = 6.
Step 1: Differentiate using the Product Rule: (1)(y) + x(dy/dx) = 0.
Step 2: Solve: x(dy/dx) = −y, so dy/dx = −y/x.
Example 3: More Complex Equation
Find dy/dx for x³ + y³ = 6xy.
Step 1: Differentiate: 3x² + 3y²(dy/dx) = 6y + 6x(dy/dx).
Step 2: Collect dy/dx terms: 3y²(dy/dx) − 6x(dy/dx) = 6y − 3x².
Step 3: Factor: (3y² − 6x)(dy/dx) = 6y − 3x².
Step 4: Solve: dy/dx = (6y − 3x²) / (3y² − 6x) = (2y − x²) / (y² − 2x).
Tangent Lines to Implicit Curves
Example 4: Tangent to an Ellipse
Find the equation of the tangent line to x²/4 + y²/9 = 1 at the point (1, 3√3/2).
Step 1: Differentiate: 2x/4 + 2y(dy/dx)/9 = 0, i.e., x/2 + (2y/9)(dy/dx) = 0.
Step 2: Solve: dy/dx = −9x/(4y).
Step 3: At (1, 3√3/2): dy/dx = −9(1)/(4 · 3√3/2) = −9/(6√3) = −3/(2√3) = −√3/2.
Step 4: Tangent line: y − 3√3/2 = (−√3/2)(x − 1).
Second Derivatives Implicitly
Example 5: Finding d²y/dx²
Given x² + y² = 25 and dy/dx = −x/y, find d²y/dx².
Differentiate dy/dx = −x/y using the Quotient Rule:
d²y/dx² = −[y(1) − x(dy/dx)] / y²
Substitute dy/dx = −x/y:
= −[y − x(−x/y)] / y² = −[y + x²/y] / y² = −(y² + x²) / y³
Since x² + y² = 25: d²y/dx² = −25/y³.
Key Takeaways
- Implicit differentiation uses the Chain Rule: every time you differentiate y, attach a factor of dy/dx.
- The result for dy/dx may contain both x and y. That is expected.
- You can find tangent lines and second derivatives for implicit curves.
- The technique works for any equation relating x and y, including circles, ellipses, and more exotic curves.
Check Your Understanding
1. Find dy/dx for x² − y² = 16.
2. Find dy/dx for sin y + x = 1.
3. Find the slope of the tangent to x² + y² = 25 at (3, −4).
4. Find dy/dx for ey + xy = 5.
Ready for More?
Practice Problems
Test all the differentiation rules from this module with 10 practice problems.
Practice Problems