x² + y² = 25 cannot be written as a single function of x. Implicit differentiation handles it without solving for y.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The method
Differentiate both sides with respect to x, treating y as a function of x throughout.
Every y term needs the Chain Rule
Differentiating y² gives 2y · dy/dx. The dy/dx factor is the inner derivative, and forgetting it is the classic error.
Then solve
Collect every term containing dy/dx on one side, factor it out, and divide.
The answer holds both variables
An implicit derivative usually depends on x and y together. That is expected, not a mistake.
Tangent lines
Substitute both coordinates of the point into dy/dx to get the slope, then use point-slope form.
Differentiating an equation you cannot solve
For x² + y² = 25, solving for y gives two functions and awkward algebra. Differentiating both sides directly, treating y as a function of x, avoids that entirely.
Every y picks up a dy/dx
Differentiating y² gives 2y·dy/dx, because y depends on x. That extra factor is the chain rule, and omitting it is the defining error of implicit differentiation.
Then solve for dy/dx
Collect all terms containing dy/dx on one side, factor it out and divide. The answer usually involves both x and y, which is expected — the slope depends on which point on the curve you are at.
Where it is needed
Curves that are not functions, related rates problems, and deriving the derivatives of inverse functions such as arcsin. Implicit differentiation is the tool for anything defined by a relation rather than a formula.
Step 2: Try It Yourself
Tap and try it out.
- Diameter10 cm
- Circumference31.42 cm
- Circumference ÷ diameter3.14
Change the size. The circle gets bigger, but circumference divided by diameter stays at about 3.14 every time. That number is π.
Step 3: Watch an Example
One step at a time.
Watch Elena Differentiate a Circle
Elena finds dy/dx for x² + y² = 25.
- Step 1
Differentiating x² gives 2x.
Step 4: Your Turn
Practice makes it stick.
The Slope
Problem 1 of 2
For x² + y² = 25, dy/dx = −x ÷ y. What is the slope at (3, 4), as a decimal?
The Other Point
Problem 2 of 2
Same curve. What is the slope at (4, 3), as a decimal to two places?
Differentiate Implicitly
1 of 8
Differentiating y² with respect to x gives 2y times what?
2 of 8
For x² + y² = 25, slope at (0, 5)?
3 of 8
For x² + y² = 25, slope at (3, −4), as a decimal?
4 of 8
x² + y² = 169. Is (5, 12) on the curve? 1 yes, 0 no.
5 of 8
For x² + y² = 169, slope at (5, 12), as a decimal to two places?
6 of 8
What is the derivative of a constant?
7 of 8
Put the implicit differentiation process in order.
- 1Attach dy/dx to every term that came from a y.
- 2Collect the dy/dx terms on one side.
- 3Factor out dy/dx and divide.
- 4Differentiate both sides with respect to x.
8 of 8
For x² + y² = 100, slope at (6, 8), as a decimal?
Step 5: Quick Check
Show what you know.
Question 1 of 2
For x² + y² = 25, what is the slope at (4, 3), as a decimal to two places?
Question 2 of 2
Why does differentiating y² give 2y · dy/dx?
What You Learned
- 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.