AcademyDifferentiation
Academy
Implicit Differentiation
Level 1 - Math I (Physics) topic page in Differentiation.
Implicit Differentiation
Implicit differentiation is used when \(y\) is not explicitly solved as a function of \(x\).
Method
- Differentiate both sides with respect to \(x\)
- Treat \(y\) as a function of \(x\): \(\frac{d}{dx}[y] = \frac{dy}{dx}\)
- Collect \(\frac{dy}{dx}\) terms on one side
- Solve for \(\frac{dy}{dx}\)
Example: Circle
\(x^2 + y^2 = r^2\)Differentiating:
Step 1
\[2x + 2y\frac{dy}{dx} = 0\]
Result
\[\frac{dy}{dx} = -\frac{x}{y}\]
Example: Ellipse
\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)Ellipse Derivative
\[\frac{dy}{dx} = -\frac{b^2x}{a^2y}\]