Either equation format
Enter both sides of an equation or a single expression understood as equal to zero.
Implicit Differentiation Calculator with Steps
Enter an equation that relates two variables. The calculator rewrites it as F = 0, finds both partial derivatives, solves for the dependent derivative, and checks an optional point—all locally in your browser.
Enter an implicit equation
Use ^ for powers and * for multiplication. Supported functions include sin, cos, tan, exp, ln, log, sqrt, and abs.
The default finds dy/dx. Choose different variables to find another first derivative.
Assign every variable needed. The point is checked against the equation.
Try an example
Keyboard shortcut: Ctrl/⌘ + Enter.
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Implicit derivative
Equation in standard form
Enter an equation to begin.
Result
The exact implicit derivative will appear here.
Step-by-step implicit differentiation
Rule-by-rule working will appear here.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Rewrite as F = 0 → calculate Fx and Fy → apply dy/dx = −Fx/Fy → simplify.
- Input limits
- One equation up to 280 characters, single-letter variables, and first implicit derivatives.
- Privacy
- Your equation and point values stay in your browser; no solving API receives them.
- Domain note
- The formula is locally valid where the equation is differentiable and the dependent partial derivative is nonzero.
How to use the implicit differentiation calculator
- Enter an equation such as
x^2+y^2=25, or enterx^2+y^2-25to mean F(x,y) = 0. - Choose the independent and dependent variables. Keep x and y selected to calculate
dy/dx. - Optionally enter a point such as
x=3, y=4. The calculator verifies that the point is on the curve before reporting its slope. - Select Calculate derivative, then review, copy, or download the work.
The implicit differentiation formula
Write the equation as F(x,y) = 0 and regard y as a differentiable function of x. The multivariable chain rule gives:
\[\frac{d}{dx}F(x,y(x))=F_x+F_y\frac{dy}{dx}=0\qquad\Longrightarrow\qquad\frac{dy}{dx}=-\frac{F_x}{F_y}\]
This rearrangement requires Fy ≠ 0. If Fy = 0 but Fx ≠ 0 at a point, the curve may instead have a vertical tangent there. When both partial derivatives vanish, higher-order analysis may be needed.
What the calculator handles
Standard derivative rules
Power, product, quotient, chain, trigonometric, exponential, logarithmic, root, and absolute-value rules are supported.
Variable selection
Calculate dy/dx, dx/dy, or a derivative involving t by choosing independent and dependent variables.
Point checks
Point evaluation checks curve membership, finite slope, vertical-tangent candidates, and singular cases.
Worked implicit differentiation examples
Circle
x²+y²=25
Differentiating gives 2x + 2y·y′ = 0, so y′ = −x/y. At (3,4), the slope is −3/4.
Product curve
xy=6
The product rule gives y + x·y′ = 0, so y′ = −y/x.
Quadratic relation
x²+xy+y²=7
Collecting y′ terms gives 2x + y + (x + 2y)y′ = 0, so y′ = −(2x+y)/(x+2y).
Implicit differentiation calculator FAQs
What is implicit differentiation?
It finds a derivative when the dependent variable appears inside an equation rather than being isolated. Because y depends on x, differentiating a term involving y introduces dy/dx through the chain rule.
Why does the formula use partial derivatives?
For F(x,y) = 0, Fx describes the direct change with x while y is held constant, and Fy multiplies the indirect change dy/dx. Their total must remain zero along the level curve.
Can I enter an expression without an equals sign?
Yes. The calculator treats the expression as F = 0. For example, x^2+y^2-25 and x^2+y^2=25 describe the same curve.
What happens at a vertical tangent?
When Fy = 0 and Fx ≠ 0 at a point on the curve, dy/dx is undefined and the point is a vertical-tangent candidate. Switching the variables may produce a finite dx/dy.
Why might a point result be inconclusive?
If both Fx and Fy are zero, the usual first-derivative formula becomes 0/0. The curve may cross itself, have a cusp, contain an isolated point, or require higher-order analysis.
Can I use parameters?
Yes. Single-letter symbols such as r or a can remain as parameters. Assign them in the optional point field when a numerical slope requires their values.
Are trigonometric inputs in radians?
Yes. Standard calculus derivative formulas use radian measure.
Does my equation leave the browser?
No. The symbolic calculation and point evaluation run on your device.