# Find dy/dx by implicit differentiation

Finds dy/dx for an equation in x and y, such as x^2 + y^2 = 25, and the slope at a point on the curve, checked numerically.

- Page: https://www.acalculator.org/math/implicit-differentiation-calculator
- JSON spec: https://www.acalculator.org/math/implicit-differentiation-calculator.json
- Version: 0f5fd6a41854

## Default answer

Example with the default inputs (Equation in x and y x^2 + y^2 = 25): For x^2 + y^2 = 25, dy/dx = -x/y.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| eq | Equation in x and y | An equation such as x^2 + y^2 = 25 or x^3 sin(y) + y = 4x + 3. |
| at | Slope at (x, y) (optional) | A point on the curve, typed like 3, -4. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| derivative | dy/dx = | The derivative of y with respect to x, in terms of x and y. |
| slope | Slope at the point | dy/dx at the given point of the curve. |

## Method

Write the equation as H(x, y) = 0 (left side minus right side). Then dy/dx = −(∂H/∂x)/(∂H/∂y), checked against difference quotients of H.

## Assumptions

- y is a differentiable function of x near each point; angles are in radians; ln is the natural logarithm.
- An answer that fails its check is not shown.

## Worked examples

1. eq = x^2 + y^2 = 25, at = 3, -4 gives derivative = -x/y, slope = 0.75. Source: OpenStax, Calculus Volume 1, section 3.8 Implicit Differentiation, Example 3.68 and 3.71. https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation.
2. eq = x^3 sin(y) + y = 4x + 3 gives derivative = (4 - 3x^2 sin(y))/(x^3 cos(y) + 1).
3. eq = y^3 + x^3 - 3x y = 0, at = 3/2, 3/2 gives derivative = (y - x^2)/(y^2 - x), slope = -1.

## FAQ

### What is implicit differentiation?

A way to find dy/dx when y is not written as a function of x, as in x² + y² = 25. Differentiate both sides with respect to x, using the chain rule on every term with y (the derivative of y² is 2y · dy/dx), then solve for dy/dx.

### How do I differentiate x² + y² = 25?

2x + 2y · dy/dx = 0, so dy/dx = −x/y. At the point (3, −4) on the circle, the slope is −3/(−4) = 3/4.

### Is there a formula?

Yes. Move everything to one side, H(x, y) = 0. Then dy/dx = −H_x/H_y, where H_x is the derivative of H in x with y held fixed and H_y the derivative in y with x held fixed. It gives the same answer as differentiating term by term.

### Why is the answer in terms of both x and y?

Because y is only defined implicitly: at a given x there may be several points on the curve (the circle has two at x = 3), each with its own slope. Give a point (x, y) to get one number.

### Why does the page say the point is not on the curve?

The slope formula only makes sense at points that satisfy the equation. The page checks that the two sides agree at your point to 1 part in 10⁹. For x² + y² = 25, (3, 4) and (3, −4) are on the curve, but (1, 1) is not.

### How is the answer checked?

At test points the formula for dy/dx is compared with −(difference quotient of H in x)/(difference quotient of H in y). They must agree at every test point where both are real numbers, and at least 3 such points.

## Sources

- OpenStax, Calculus Volume 1, section 3.8 Implicit Differentiation: https://openstax.org/books/calculus-volume-1/pages/3-8-implicit-differentiation