Equations in x and y · Solve for dy/dx

Implicit Differentiation Calculator

Enter an expression ↓

Enter an equation containing x and y. Differentiate both sides, treat y as a function of x, and isolate dy/dx with a deterministic partial-derivative identity.

Equation-aware parserdy/dx shown explicitlyNo AI-generated answer

Dedicated workspace

Implicit differentiation

Engine

Use ^ for powers and parentheses for functions. You can enter sec(x), csc(x), cot(x) and arctan(x) directly; trigonometric inputs use radians.

Input help and supported methods

Keep the original question visible. Confirm the variable and the live preview before calculating. Use ^(1/2) for a square root and explicit parentheses for function arguments.

Enter both sides of the equation with one equals sign. Choose distinct independent and dependent variables. We never invent the missing right-hand side.

Read the method

Try one. See the steps.

Click to fill the input. Calculation starts only when you choose Solve.

Live problem preview

If y is already written explicitly as y=f(x), use the ordinary derivative calculator. This page handles equations where x and y are linked implicitly. All derivative tools

Differentiate a relation

What is implicit differentiation?

Implicit differentiation finds a derivative when y is defined by an equation instead of being isolated as y=f(x). Every y term must carry its hidden dependence on x.

Why dy/dx appears beside y

In x²+y²=25, y changes with x along the circle. Differentiating y² therefore gives 2y·dy/dx through the chain rule, not simply 2y.

The differentiated equation is 2x+2y·dy/dx=0. Moving 2x and dividing by 2y produces dy/dx=−x/y wherever y is nonzero.

Organize the work with partial derivatives

Move an equation into F(x,y)=0. Its total derivative is Fₓ+Fᵧ·dy/dx=0, so dy/dx=−Fₓ/Fᵧ when the denominator is nonzero.

The calculator parses both sides, forms one relation, computes the two partial derivatives with the deterministic engine, and displays their negative ratio.

Products containing x and y

For x²+xy+y²=1, the product rule changes xy into y+x·dy/dx. The other derivative terms are 2x and 2y·dy/dx.

Collecting derivative terms gives (x+2y)dy/dx=−(2x+y), so the slope is −(2x+y)/(x+2y).

Interpret an undefined derivative

When Fᵧ is zero, dy/dx may be undefined, a vertical tangent may occur, or the point may be singular. A zero denominator is mathematical information.

Check that any point substituted into the derivative also satisfies the original equation. The symbolic slope formula only applies along the stated relation.

A reliable way to find dy/dx implicitly

Begin by checking that the input is an equation, not an isolated expression. An implicit relation such as x²+y²=25 describes many ordered pairs at once. Although y is not written as y(x), its value changes as x moves along the curve. That hidden dependence is why derivatives of y-terms produce dy/dx. Constants such as 25 or r², when r is fixed, differentiate to zero. Differentiate every term on both sides with respect to x. Pure x terms follow ordinary rules. Pure y terms use the chain rule: d/dx(y²)=2y·dy/dx and d/dx(sin y)=cos(y)·dy/dx. A mixed term such as xy needs the product rule because both factors can change along the curve. Its derivative is y+x·dy/dx. Writing the derivative marker beside every affected y-term prevents the most common mistake, which is treating y as a constant. After differentiating, collect every term containing dy/dx on one side. Move the remaining terms to the other side, factor out dy/dx, and divide. For x²+xy+y²=1, differentiation gives 2x+y+x·dy/dx+2y·dy/dx=0. Grouping produces (x+2y)dy/dx=−(2x+y), and division gives dy/dx=−(2x+y)/(x+2y). This is the same result produced by the compact identity dy/dx=−Fₓ/Fᵧ after the equation is written as F(x,y)=0. The partial-derivative identity is a clean computational route, but it does not remove the underlying chain rule. Fₓ measures how the relation changes when x moves while y is held fixed. Fᵧ measures the y-direction change. Along the curve, their changes must balance, which leads to Fₓ+Fᵧ·dy/dx=0. CalculusMate displays the relation, the two partial derivatives, and their negative ratio so the reasoning remains reviewable. A symbolic derivative is not automatically a numerical slope. To find the slope at a point, first verify that the point satisfies the original equation, then substitute both coordinates into dy/dx. For the circle x²+y²=25 at (3,4), the slope is −3/4. A point not on the circle has no meaning for that curve, even if it can be inserted into the formula. If the denominator is zero, investigate a vertical tangent or singular point rather than reporting an ordinary finite slope. Some relations contain several branches. Solving a circle explicitly gives an upper and lower branch, yet implicit differentiation handles both with one formula. The coordinate values determine the sign. This is one reason implicit differentiation is useful: it avoids unnecessary algebra and can work even when isolating y is difficult. Still, the method assumes the relation is differentiable near the point and that the denominator used to isolate dy/dx is nonzero. For clear written work, preserve the original equation, show the derivative of each side, mark every chain-rule dy/dx factor, collect those factors, and only then simplify. The page follows that order. It accepts one supported equation and returns a deterministic symbolic slope; it does not silently solve singular curves or invent a branch when the mathematics needs additional analysis.

Check the curve as well as the algebra

Practice by choosing a point on each relation and checking the slope there. For a circle, compare opposite points to see how the signs change. For xy=c, notice that the derivative −y/x agrees with the slope of y=c/x wherever x is nonzero. These comparisons connect the implicit method to familiar explicit functions without requiring you to isolate y every time. When reviewing an answer, count every original y occurrence and confirm that its derivative produced a dy/dx factor, including y hidden inside a product or function. Then verify that all derivative factors were collected before division. If a requested point makes the denominator zero, return to the original curve and analyze its local geometry. A calculator can expose that condition, but deciding whether it represents a vertical tangent, a cusp, or a singular crossing needs the equation and the point together.

Equations you can test

Watch hidden derivatives become visible

x2+y2=25x^{2} + y^{2} = 25

The circle gives dy/dx=−x/y wherever y is not zero.

x²+xy+y²=1

The product xy requires the product rule before dy/dx terms can be collected.

xy=6

Differentiate the product to obtain y+x·dy/dx=0, then solve as dy/dx=−y/x.

Supported equation boundaries

Enter exactly one equation using supported arithmetic and functions. The solver returns a symbolic derivative; it does not solve the original curve for y or classify singular points.

FAQ
How do I enter an implicit equation?+

Enter one equation such as x^2+y^2=25. Use ^ for powers and * where multiplication may be unclear.

Why is y treated as a function of x?+

Along the curve, changing x generally changes y. The chain rule records that dependence with dy/dx.

Can it find a slope at a point?+

It returns symbolic dy/dx. Substitute a valid point from the original equation for a numerical slope.

What if the denominator is zero?+

The slope may be undefined or the curve may have a vertical tangent or singular point.

Related calculators

Check the result and its conditions

CalculusMate applies deterministic rules to supported derivatives, integrals and limits. Read the result together with its variable, domain and verification status. A rule that is not supported does not make the mathematical problem impossible.

  1. Enter the expression and choose the variable; provide bounds only for a definite integral.
  2. Check domain restrictions and sign intervals. In real arithmetic, sqrt(g²)=|g|, so the absolute value cannot simply be removed.
  3. Read the calculation steps and status. A numerical approximation is different from an independently checked symbolic result.
Three reference examples from the main calculators
TaskExpressionAnswerConditions
Derivativex3x^{3}3 x23\,x^{2}Real x
Indefinite integral2 x2\,xx2+Cx^{2} + CC is an arbitrary constant
Limit as x → 0sin⁡xx\frac{\sin x}{x}11Radians; x approaches 0, not substitution at 0

Mathematical references

OpenStax explains the differentiation rules and the Fundamental Theorem of Calculus used in the examples. These textbook references describe the mathematics; they do not certify this software or its coverage.

Maintained by CalculusMate · Guide updated: · Calculation and verification methodology