What is implicit differentiation?
Differentiating an equation that defines y implicitly as a function of x.
Implicit Differentiation Calculator
Differentiate F(x,y)=constant implicitly to solve for y′, then evaluate at (x₀,y₀).
Choose the relation
Select circle or ellipse form, enter parameters, and a point on the curve.
About this calculator
How to use Implicit Differentiation Calculator, plus the formulas and logic behind the results.
Tool description
Implicit differentiation treats y as a function of x and differentiates both sides of F(x,y)=k.
How to use
- Select the circle or ellipse equation.
- Enter radius or ellipse coefficients.
- Enter a point (x₀, y₀) on the curve.
- Select Calculate to view dy/dx formula and value.
Formulas & reference
Circle: 2x + 2y y′ = 0 → y′ = −x/yEllipse: 2ax + 2by y′ = 0 → y′ = −(ax)/(by)
Implicit tips
- Verify the point satisfies the original equation
- y = 0 on a circle gives a vertical tangent
- Same idea extends to more complicated F(x,y)
- Result is the slope of the tangent line
Calculation logic
Applies implicit differentiation, reports the symbolic slope formula, and evaluates at the given point if y₀ ≠ 0.
Vertical tangents occur where the denominator is zero; the point should lie on the curve.
Implicit Differentiation Calculator FAQ
Common questions about this calculator and how to use the results.
Why divide by y?
The chain rule produces a y′ term multiplied by the y-partial.
What if y₀ = 0?
The tangent is vertical (undefined slope) for these standard forms.
Must the point be on the curve?
Yes for a geometric tangent; the tool also reports the residual.
Ellipse vs circle?
A circle is the special case a = b with r² = c/a.
Can I get the tangent line?
Use y − y₀ = y′(x − x₀) with the computed slope.