AI Implicit Functions

AI Differentiation Solver for Implicit Functions

\[
\frac{d}{dx}[y] = \frac{dy}{dx}
\]

- Polynomial Equations
Such as \( x^2 + y^2 = 25 \), where terms involving \( y \) and \( x \) are combined.
- Equations with Trigonometric Functions
For example, \( \sin(x + y) = x \cdot y \), where trigonometric relationships complicate isolation of variables.
- Logarithmic and Exponential Equations
Equations like \( x \cdot e^y + y \cdot \ln(x) = 10 \), requiring implicit differentiation with chain rule applications.
- Mixed and Multivariable Equations
Complex equations involving mixed terms and partial derivatives, applicable in physics, engineering, and multivariable calculus.

Input the Implicit Function
Users simply enter the equation, and the solver’s algorithms parse it for terms involving \( y \) as a function of \( x \).

Apply Implicit Differentiation Rules
The AI applies chain rules, product rules, and other necessary differentiation techniques automatically.

Step-by-Step Output
The solution is displayed in detailed steps, breaking down each part of the process for user understanding.

What types of equations are supported?
Explain that polynomial, trigonometric, exponential, and multivariable equations are all supported.

Is this tool suitable for advanced calculus problems?
Reassure users that the solver is built for both basic and advanced implicit differentiation needs.

Can I see the solution steps?
Confirm that all steps are shown for an educational experience.

Why Use Implicit Differentiation?

Implicit differentiation is essential for differentiating equations where it is difficult or impossible to isolate \( y \) in terms of \( x \). It allows us to find the derivative \( \frac{dy}{dx} \) directly without solving for \( y \).

Chain Rule in Implicit Differentiation

When differentiating with respect to \( x \), remember that \( y \) is a function of \( x \) (i.e., \( y = f(x) \)). Therefore, when you differentiate any term involving \( y \) with respect to \( x \), you must apply the chain rule:
\[
\frac{d}{dx}[y] = \frac{dy}{dx}
\]
For instance, if we differentiate \( y^2 \) with respect to \( x \), we get:
\[
\frac{d}{dx}[y^2] = 2y \cdot \frac{dy}{dx}
\]

Step-by-Step Method for Implicit Functions

To differentiate an implicit equation with respect to \( x \)

1. Differentiate both sides of the equation with respect to \( x \), applying the chain rule to any term involving \( y \).

2. Isolate \( \frac{dy}{dx} \) on one side of the equation.

3. Solve for \( \frac{dy}{dx} \) to find the derivative in terms of \( x \) and \( y \).

Basic Implicit Differentiation

Given:
\[
x^2 + y^2 = 25
\]
Differentiate both sides with respect to \( x \):
\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)
\]
Applying the derivatives:
\[
2x + 2y \cdot \frac{dy}{dx} = 0
\]
Solve for \( \frac{dy}{dx} \):
\[
2y \cdot \frac{dy}{dx} = -2x
\]
\[
\frac{dy}{dx} = -\frac{x}{y}
\]
Result: The derivative \( \frac{dy}{dx} = -\frac{x}{y} \).

Finding the Tangent Line at a Point

Given

\[
x^3 + y^3 = 6xy
\]
To find the slope of the tangent at a specific point, say \( (1,2) \):
1. Differentiate both sides with respect to \( x \):
\[
3x^2 + 3y^2 \frac{dy}{dx} = 6 \frac{dy}{dx} \cdot y + 6x
\]
2. Collect terms involving \( \frac{dy}{dx} \) and solve for it.
3. Substitute \( (1,2) \) to get the specific slope and equation of the tangent line.