How to use: Enter each term of the equation f(x, y) = [RHS] in the form coeff*x^a*y^b. For example, x² + y² = 25 uses two terms: (1, x^2, y^0) and (1, x^0, y^2) with RHS = 25.
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Coefficient
x exponent
y exponent
Implicit Derivative Calculator: How the Method Works
For an equation f(x, y) = c, differentiating both sides with respect to x gives: df/dx + df/dy * (dy/dx) = 0. Solving for dy/dx yields the formula below. This works for any differentiable implicit equation.
dy/dx = -(df/dx) / (df/dy) For term: c x x^a x y^b df/dx = c x a x x^(a-1) x y^b df/dy = c x b x x^a x y^(b-1)
The formula dy/dx = -(df/dx) / (df/dy) directly uses partial derivatives of f with respect to each variable. This is the same operation computed by the Partial Derivative Calculator, which finds df/dx and df/dy separately for any polynomial in x and y.
Second Derivative Implicit Differentiation: Finding d²y/dx²
To find the second derivative d²y/dx² of an implicitly defined function, differentiate dy/dx with respect to x. Since dy/dx = -(df/dx) / (df/dy), apply the quotient rule to this expression, then substitute your earlier expression for dy/dx.
The key step is substituting the expression for dy/dx back into the result after applying the quotient rule. This eliminates dy/dx from the second derivative expression and leaves d²y/dx² in terms of x and y only. For the reverse operation of differentiation, see the Antiderivative Calculator.
Implicit Differentiation and Partial Derivatives
Implicit differentiation and partial derivatives are deeply connected. The formula dy/dx = -(df/dx) / (df/dy) can be read as: the ratio of the partial derivative of f with respect to x, divided by the partial derivative of f with respect to y, negated.
Operation
What it computes
Notation
Key rule applied
Partial df/dx
Rate of change w.r.t. x, y held constant
df/dx or f_x
Power rule, y treated as constant
Partial df/dy
Rate of change w.r.t. y, x held constant
df/dy or f_y
Power rule, x treated as constant
Implicit dy/dx
How y changes as x changes on the curve
dy/dx
Chain rule, dy/dx factor on y terms
Relationship
dy/dx = -(df/dx) / (df/dy)
Implicit Function Theorem
Both operations used together
The Implicit Function Theorem guarantees that as long as df/dy is not zero at a point, the implicit equation F(x, y) = c locally defines y as a differentiable function of x near that point. For polynomial estimation between known data points (a related numerical technique), see the Interpolation Calculator.
The slope of the tangent to the circle x² + y² = 25 at point (3, 4) is -0.75. This is consistent with the geometric fact that the tangent is perpendicular to the radius from the origin to (3, 4), which has slope 4/3.
Common Mistakes to Avoid
Forgetting dy/dx when differentiating y terms
d/dx[y^2] = 2y times dy/dx, not just 2y. Every y term that you differentiate with respect to x picks up a dy/dx factor via the chain rule.
Treating y as a constant
In implicit differentiation, y is a function of x, not a fixed number. Treating it as a constant removes the chain rule factor and gives a wrong derivative.
Stopping before isolating dy/dx
After differentiating both sides, you must collect all dy/dx terms on one side and factor them out. Not doing this leaves the answer incomplete.
Using a point not on the curve
If the evaluation point (x, y) does not satisfy the original equation, the computed dy/dx is geometrically meaningless. Verify the point satisfies the equation first.
Confusing df/dy = 0 with a zero slope
If df/dy = 0 at a point, the denominator of -df/dx / df/dy is zero, making dy/dx undefined. This means the tangent line is vertical, not that the slope is zero.
Frequently Asked Questions
Implicit differentiation is a technique for finding dy/dx when y is not explicitly defined as a function of x. In an explicit function like y = x^2 + 3, you differentiate directly. In an implicit equation like x^2 + y^2 = 25, y is defined implicitly. You differentiate both sides with respect to x, applying the chain rule wherever y appears (each y term picks up a dy/dx factor), then solve for dy/dx.
Open lecture notes on implicit differentiation, the chain rule in the implicit context, and related rates applications.
HR
Hassaan Rasheed
Developer and Researcher, CalculatorFlux
Researches and verifies the formulas, methodology, and source data behind each calculator on CalculatorFlux. All tools are built and checked against the cited references before publication.
Last updated: May 2026
Common Examples
x^2 + y^2 = 25
Terms: (1,2,0) + (1,0,2)
dy/dx = -x/y
x^2 + xy + y^2 = 7
Terms: (1,2,0) + (1,1,1) + (1,0,2)
dy/dx = -(2x+y)/(x+2y)
x^3 + y^3 = 8
Terms: (1,3,0) + (1,0,3)
dy/dx = -x^2/y^2
Key Rule
When differentiating a term like y^n with respect to x, apply the chain rule: d/dx[y^n] = n times y^(n-1) times dy/dx. The dy/dx factor always appears when differentiating any function of y with respect to x.