Step by Step
1
When you need this technique
Use implicit differentiation whenever y cannot be easily isolated on one side of the equation — for example, a circle equation like x² + y² = 25, where solving explicitly for y would require splitting into two separate square-root equations.
2
The key idea: treat y as a hidden function of x
Even though y isn't written as f(x) explicitly, you treat it as if it secretly is one. That means every time you differentiate a y-term, you must apply the chain rule, which introduces a factor of dy/dx.
3
Differentiate both sides
Differentiate every term on both sides of the equation with respect to x, remembering the dy/dx factor on any y-term.
4
Collect and solve for dy/dx
Gather every term containing dy/dx onto one side of the equation, factor dy/dx out, and divide to solve for it explicitly.
Applied Walkthrough
1
Find dy/dx for the circle x² + y² = 25. Differentiate both sides with respect to x: d/dx[x²] + d/dx[y²] = d/dx[25].
2
The x² term gives 2x normally. The y² term needs the chain rule, since y is being treated as a function of x: d/dx[y²] = 2y·(dy/dx). The right side, a constant, gives 0.
3
So the equation becomes: 2x + 2y·(dy/dx) = 0.
4
Collect and solve: 2y·(dy/dx) = −2x, so dy/dx = −x/y.
Exam Application
Exams test whether you remember to attach the dy/dx factor to every y-term (via the chain rule), and whether you can correctly collect and solve for dy/dx at the end.
⚠ Common Trap
The most common trap is forgetting to multiply y-terms by dy/dx — treating y like an ordinary constant or x-term instead of recognizing it needs the chain rule applied, since y is implicitly a function of x.
✓ Quick Self-Check
1. When should you use implicit differentiation instead of solving for y first?
When y cannot be easily isolated on one side of the equation.
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2. What factor gets attached to every y-term when differentiating implicitly?
dy/dx, due to the chain rule.
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3. Find dy/dx for x² + y² = 25.
dy/dx = −x/y.
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4. What is the first step in implicit differentiation?
Differentiate both sides of the equation with respect to x.
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5. After differentiating, what do you do to isolate dy/dx?
Collect all dy/dx terms on one side, factor dy/dx out, and divide to solve.
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