01 Key Concepts
Explicit vs. Implicit
Explicit: y is isolated on one side (y = x^2). Implicit: x and y are mixed (x^2 + y^2 = 25).
Differentiate Both Sides
Take d/dx of every term on both sides of the equation, remembering that y is a function of x.
The Key Trick
Whenever you differentiate a term with y, use the chain rule and multiply by dy/dx, since y itself depends on x.
Solving for dy/dx
After differentiating, collect all dy/dx terms on one side and solve algebraically.
Where It's Used
Useful for curves like circles, ellipses, and other relations that are not functions in the simple y=f(x) sense.
02 Key Formulas
- d/dx[y^n] = n*y^(n-1) * dy/dx
03 Solved Examples
Example 1 Find dy/dx for x^2 + y^2 = 25.
- Differentiate both sides with respect to x: 2x + 2y*(dy/dx) = 0.
- Isolate dy/dx: 2y*(dy/dx) = -2x.
- Divide both sides by 2y.
Answer: dy/dx = -x/y
Example 2 Find dy/dx for x*y = 10.
- Use the product rule on the left side: d/dx[x*y] = 1*y + x*(dy/dx).
- The right side's derivative is 0.
- So y + x*(dy/dx) = 0.
- Isolate dy/dx: x*(dy/dx) = -y.
Answer: dy/dx = -y/x
Example 3 Find dy/dx for y^3 + x^2 = 4y.
- Differentiate both sides: 3y^2*(dy/dx) + 2x = 4*(dy/dx).
- Collect dy/dx terms: 3y^2*(dy/dx) - 4*(dy/dx) = -2x.
- Factor: (dy/dx)(3y^2 - 4) = -2x.
- Divide by (3y^2 - 4).
Answer: dy/dx = -2x / (3y^2 - 4)
04 Practice Questions
1Find dy/dx for x^2 + y^2 = 9.
-x/y
2Find dy/dx for x^2*y = 5.
dy/dx = -2y/x
3Find dy/dx for y^2 = x.
dy/dx = 1/(2y)
4Find dy/dx for x + y^3 = 7.
dy/dx = -1/(3y^2)
5Is y = x^2 explicit or implicit?
Explicit, y is already isolated
📄 Implicit Differentiation — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.