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Definition and meaning
Implicit differentiation finds dy/dx when x and y are tied together by an equation instead of y being isolated. Differentiate both sides with respect to x and treat y as a function y(x). Every derivative of a y-expression therefore receives a factor of y'. Then collect the y' terms and solve algebraically.
Why the rule works
The extra y' factor is an application of the chain rule. For example, d(y^2)/dx=2y·dy/dx because the outer square changes with y and y changes with x. This is why differentiating x^2+y^2=25 gives 2x+2y y'=0 rather than 2x+2y=0.
A reliable hand method
Differentiate every term with respect to x, attach y' whenever a term involving y is differentiated, move all y' terms to one side, factor out y', and divide. The compact tool here differentiates expressions with respect to the selected variable; use its result to check the term-by-term derivative before solving an entire equation.
Example 1: x^2+y^2=25
Answer: y'=-x/y
- Differentiate to obtain 2x+2y y'=0.
- Solve for y' and note y≠0 for this formula.
Example 2: x*y=6
Answer: y'=-y/x
- Use the product rule on x·y.
- The derivative is y+x y'=0.
Example 3: x^3+y^3=9
Answer: y'=-x^2/y^2
- Differentiate powers on both sides.
- Divide 3x^2+3y^2y'=0 by 3y^2.
Example 4: sin(y)=x
Answer: y'=1/cos(y)
- Chain rule gives cos(y)y'=1.
- The slope formula fails where cos(y)=0.
Common mistakes
Never treat y as a constant when differentiating with respect to x. Do not forget the product rule in terms such as xy. After solving, check whether division introduced restrictions. A vertical tangent may correspond to a denominator of zero rather than an algebra error.
How to verify an answer
Differentiate the expression independently, compare the structure of each term, and substitute a safe numerical value into both the symbolic derivative and a centered difference quotient. Avoid points outside the original domain and points near a discontinuity. A matching numerical value increases confidence, but it cannot establish equality for every input. Algebraic reasoning and domain analysis remain necessary.
When the calculator and handwritten work differ, first check parentheses, multiplication symbols, the selected variable, and any missing chain-rule factor. Then factor or expand both answers to see whether they are equivalent. The DerivCalc methodology explains why simplification is deliberately conservative.
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Questions about this calculator
Can I use the Implicit Differentiation Calculator for homework checks?
Yes. Compare the result and named rules with your own work, then check domain restrictions. The numerical spot-check is a consistency test, not a formal proof.
Why can an equivalent answer look different?
Factoring, expanding, reciprocal notation, and trigonometric identities can produce different-looking expressions with the same values on their shared domain.
Does the calculator replace learning the rule?
No. It is designed to expose the rule and intermediate reasoning. The worked examples explain how to select and apply the method by hand.
Editorial standard: formulas and notation are checked against OpenStax Calculus Volume 1, Chapter 3. Content is written and maintained by SilverCodeLabs. Report a correction with the exact expression and expected result.