Derivative of Implicit Functions

Implicit functions are those where the dependent variable is not isolated on one side of the equation. Instead, the relationship between the variables is given in an implicit form. For example, the equation of a circle $x^2 + y^2 = r^2$ is an implicit function because $y$ is not expressed solely in terms of $x$. To differentiate implicit functions, we use a technique called implicit differentiation.

Understanding Implicit Differentiation

Implicit differentiation is a method used to find the derivative of an implicit function. It is based on the chain rule for derivatives, which states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function.

When we differentiate an implicit function with respect to $x$, we treat $y$ as a function of $x$ (i.e., $y = f(x)$) even though we do not have an explicit formula for $f(x)$. We then apply the chain rule to differentiate $y$ with respect to $x$.

Steps for Implicit Differentiation

1. Differentiate both sides of the equation with respect to $x$.
2. Whenever you differentiate a term involving $y$, multiply by $\frac{dy}{dx}$ because $y$ is a function of $x$.
3. Collect all terms involving $\frac{dy}{dx}$ on one side of the equation and move all other terms to the opposite side.
4. Solve for $\frac{dy}{dx}$ to find the derivative.

Example

Consider the circle equation $x^2 + y^2 = r^2$.

1. Differentiate both sides with respect to $x$:

$$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(r^2)$$

1. Apply the chain rule to $y^2$:

$$2x + 2y\frac{dy}{dx} = 0$$

1. Solve for $\frac{dy}{dx}$:

$$2y\frac{dy}{dx} = -2x$$ $$\frac{dy}{dx} = -\frac{x}{y}$$

Therefore, the derivative of $y$ with respect to $x$ for the circle is $-\frac{x}{y}$.

Table of Differences and Important Points

Aspect	Explicit Differentiation	Implicit Differentiation
Definition	Differentiating an explicit function where the dependent variable is isolated on one side.	Differentiating an implicit function where the dependent variable is not isolated.
Method	Directly apply differentiation rules to the explicit formula.	Apply differentiation rules treating all variables as functions of $x$, using the chain rule for terms involving $y$.
Derivative Notation	$\frac{dy}{dx}$ or $f'(x)$	$\frac{dy}{dx}$, but $y$ is treated as $y(x)$.
Complexity	Generally straightforward.	Can be more complex due to the application of the chain rule.
Solving for Derivative	Not necessary, as the derivative is directly obtained.	Often requires solving an equation to isolate $\frac{dy}{dx}$.

Formulas

The general formula for implicit differentiation when differentiating a term $y^n$ with respect to $x$ is:

$$\frac{d}{dx}(y^n) = n \cdot y^{n-1} \cdot \frac{dy}{dx}$$

Additional Example

Let's differentiate the implicit function $x^3 + y^3 = 3xy$.

1. Differentiate both sides with respect to $x$:

$$\frac{d}{dx}(x^3) + \frac{d}{dx}(y^3) = \frac{d}{dx}(3xy)$$

1. Apply the chain rule and product rule:

$$3x^2 + 3y^2\frac{dy}{dx} = 3y + 3x\frac{dy}{dx}$$

1. Collect terms involving $\frac{dy}{dx}$:

$$3y^2\frac{dy}{dx} - 3x\frac{dy}{dx} = 3y - 3x^2$$

1. Factor out $\frac{dy}{dx}$ and solve:

$$\frac{dy}{dx}(3y^2 - 3x) = 3y - 3x^2$$ $$\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x}$$

Thus, the derivative $\frac{dy}{dx}$ of the implicit function $x^3 + y^3 = 3xy$ is $\frac{3y - 3x^2}{3y^2 - 3x}$.

Implicit differentiation is a powerful tool for finding derivatives when explicit formulas are not available. It is essential for dealing with equations that define curves and shapes in geometry, as well as in many applications in physics and engineering.