含 x 与 y 的方程 · 求解 dy/dx

隐函数求导计算器

输入包含 x 与 y 的方程。对等式两边求导，把 y 视为 x 的函数，并用确定性的偏导恒等式解出 dy/dx。

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

专属计算工作台

隐函数求导

计算引擎

使用 ^ 表示乘方，函数请使用括号，例如 sin(x)、log(x) 或 exp(2x)。

试试这些例题

Differentiate a relation

什么是隐函数求导？

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

看隐藏的导数一步步出现

x²+y²=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.