Implicit Differentiation: How to Approach It
When an equation links (x) and (y) together and (y) is not isolated, differentiate both sides with respect to (x) while treating (y) as a function of (x). This means every time you differentiate a term containing (y), you must include (\frac{dy}{dx}).
Repeatable method
- Differentiate term by term. Apply the usual rules to each side.
- Use the chain rule for (y)-terms. For example, (\frac{d}{dx}(y^n)=n y^{n-1}\frac{dy}{dx}).
- Keep (\frac{dy}{dx}) terms together. Move all terms involving (\frac{dy}{dx}) to one side.
- Factor out (\frac{dy}{dx}). Solve for (\frac{dy}{dx}).
- Simplify carefully. Reduce fractions and combine like terms if possible.
Good habits
- Differentiate constants as 0.
- Remember product, quotient, and chain rules when they appear.
- If the problem includes a point, substitute it only after finding (\frac{dy}{dx}).
Check your work
A quick check is to see whether your final expression isolates (\frac{dy}{dx}) correctly and whether every differentiated (y)-term has a matching (\frac{dy}{dx}). If a point is given, plug it in and make sure the result is sensible.