Some equations cannot be solved for y before differentiating, because y is mixed in with x on both sides. Implicit differentiation differentiates the equation as it stands, treating y as a function of x. You can check your work with the free tools on Calc Solver.
When You Need It
Use implicit differentiation when y cannot be isolated easily, as in x^2 + y^2 = 25 or x^3 + y^3 = 6xy. If you can solve for y first, ordinary differentiation is simpler, but for these mixed equations, implicit differentiation is the right tool.
Step 1: Differentiate Both Sides
Differentiate every term with respect to x. The derivative of x^2 is 2x. For a y term, apply the chain rule: the derivative of y^2 is 2y times dy/dx, because y itself depends on x. That extra dy/dx factor is the whole point of the method.
Step 2: Collect the dy/dx Terms
Move every term containing dy/dx to one side of the equation and everything else to the other side. For x^2 + y^2 = 25, differentiating gives 2x + 2y dy/dx = 0, so 2y dy/dx = -2x.
Step 3: Solve for dy/dx
Divide to isolate dy/dx. From 2y dy/dx = -2x, you get dy/dx = -x/y. The answer is allowed to contain both x and y, which is normal for implicit differentiation.
Worked Example
Find dy/dx for x^3 + y^3 = 6xy. Differentiating gives 3x^2 + 3y^2 dy/dx = 6y + 6x dy/dx, using the product rule on 6xy. Collect dy/dx terms: 3y^2 dy/dx – 6x dy/dx = 6y – 3x^2. Factor and solve: dy/dx = (6y – 3x^2)/(3y^2 – 6x), which simplifies to (2y – x^2)/(y^2 – 2x).
Common Mistakes to Avoid
First, forgetting the dy/dx factor when differentiating a y term. Second, missing the product rule when x and y multiply each other. Third, expecting the answer to contain only x: an answer with y in it is correct and expected here.
Practice Problems to Try
Find dy/dx for x^2 + y^2 = 9, for x^2 – y^2 = 1, and for xy = 4. Check each answer by substituting a point on the curve if one is easy to find.