Enter an equation in x and y, such as x³ + y³ = 6xy, to get dy/dx with every step of the chain rule shown. Add a point to evaluate the slope, the tangent and normal lines and the second derivative, and see the curve drawn on a graph.
Last updated:
Examples:
Point (optional)
(,)
Use x and y with ^ for powers, sqrt, ln, e^x, sin, cos, tan and implicit multiplication (6xy). One "=" sign; without it the equation is read as … = 0. Leave the point's y empty to solve for y on the curve.
dy/dx
Curve F(x, y) = 0Tangent
Drag to pan, scroll to zoom. Click the curve to move the point there.
Step-by-step
Advertisement
Advertisement
How to use the Implicit Differentiation Calculator
Type the equation using x and y, for example x^2 + y^2 = 25 or sin(x*y) = y.
Optionally enter a point (x, y). Leave y empty to have the calculator solve for the matching y on the curve.
Press Find dy/dx (or Enter). The derivative appears as a fraction, followed by the slope and tangent line at your point.
Read the steps: each term is differentiated with respect to x, the y′ terms are collected, and the equation is solved for y′.
Explore the graph: drag to pan, scroll to zoom, and click anywhere on the curve to evaluate the slope there.
How implicit differentiation works
Differentiate both sides with respect to x. Terms with only x differentiate normally.
Apply the chain rule to y terms: d/dx[g(y)] = g′(y)·y′. For example d/dx(y³) = 3y²·y′ and d/dx(sin y) = cos y·y′.
Use the product rule for mixed terms: d/dx(xy) = y + x·y′ and d/dx(x²y) = 2xy + x²·y′.
Collect every y′ term on one side, factor out y′, and divide.
All of this is summed up by the formula dy/dx = −Fx / Fy for an equation written as F(x, y) = 0.
Worked example: the folium of Descartes
Differentiate x³ + y³ = 6xy. The left side gives 3x² + 3y²·y′. The right side needs the product rule: 6y + 6x·y′. So 3x² + 3y²·y′ = 6y + 6x·y′. Collect the y′ terms: (3y² − 6x)·y′ = 6y − 3x². Divide and cancel the common factor 3:
dy/dx = (2y − x²) / (y² − 2x), which is the same as (x² − 2y) / (2x − y²), the form the calculator prints. At (3, 3) this is (6 − 9) / (9 − 6) = −1, so the tangent there is y = −x + 6.
Common implicit derivatives
Equation
dy/dx
Note
x² + y² = r²
−x / y
Circle: the tangent is perpendicular to the radius
x²/a² + y²/b² = 1
−b²x / (a²y)
Ellipse
x²/a² − y²/b² = 1
b²x / (a²y)
Hyperbola
xy = c
−y / x
Rectangular hyperbola
y² = 4ax
2a / y
Sideways parabola
x³ + y³ = 6xy
(2y − x²) / (y² − 2x)
Folium of Descartes
Tips
Forgetting the y′ after differentiating a y term is the most common mistake; the table in the steps marks each place where it appears.
Horizontal tangents occur where the numerator is 0 (and the denominator is not); vertical tangents where the denominator is 0.
Always check that your point satisfies the original equation before plugging it in. The calculator warns you if it does not.
Frequently asked questions
What is implicit differentiation?
It is a way to find dy/dx when y is not written as an explicit function of x. You differentiate both sides of the equation with respect to x, treating y as a function of x, so every derivative of a y term picks up a factor dy/dx (the chain rule). Then you solve the result for dy/dx.
How do you implicitly differentiate x² + y² = 25?
Differentiate each term: d/dx(x²) = 2x, d/dx(y²) = 2y·dy/dx, and d/dx(25) = 0. That gives 2x + 2y·dy/dx = 0, so dy/dx = −x/y. At the point (3, 4) the slope is −3/4.
Is there a shortcut formula?
Yes. Write the equation as F(x, y) = 0. Then dy/dx = −F_x / F_y, where F_x and F_y are the partial derivatives of F with respect to x and y. This is exactly what the collected terms in the step-by-step solution show.
Why does the answer contain both x and y?
Because the slope depends on where you are on the curve, and for an implicit curve a single x can belong to several points. On the circle x² + y² = 25, x = 3 belongs to (3, 4) with slope −3/4 and to (3, −4) with slope 3/4. Plug in both coordinates of a point to get a number.
How do I find the second derivative implicitly?
Differentiate dy/dx again with respect to x, still treating y as a function of x, and substitute the first derivative wherever y′ appears. Equivalently, y″ = −(F_xx·F_y² − 2F_xy·F_x·F_y + F_yy·F_x²) / F_y³. The calculator evaluates this at your point.
What does a vertical tangent or a singular point mean?
If F_y = 0 but F_x ≠ 0 at a point, the slope is undefined and the tangent line is vertical, such as the circle at (5, 0). If both F_x and F_y are 0, the point is singular (for example where a curve crosses itself), and no single slope exists.
Which functions are supported?
Polynomials, products and quotients of x and y, powers and roots, e^x and other exponentials, ln and log, trigonometric, inverse trigonometric and hyperbolic functions, and abs. Implicit multiplication like 6xy or 2x·y is understood.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.