Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly, meaning that it is not explicitly solved for one variable in terms of the other.
The Basics
Let’s consider an equation relating $ x $ and $ y $ , such as:
$$ x^2 + y^2 = 25 $$ We can’t easily solve this equation for $ y $ in terms of $ x $ . However, we can still find the derivative $ \frac{dy}{dx} $ by using implicit differentiation.
The key idea is to differentiate both sides of the equation with respect to $ x $ , treating $ y $ as a function of $ x $ . This means we’ll use the chain rule whenever we encounter a term involving $ y $ .
Steps for Implicit Differentiation
- Differentiate both sides of the equation with respect to $ x $ . Remember to use the chain rule when differentiating terms involving $ y $ .
- Solve the resulting equation for $ \frac{dy}{dx} $ . This may involve algebraic manipulation.
Example
Let’s find $ \frac{dy}{dx} $ for the equation $ x^2 + y^2 = 25 $ .
- Differentiate both sides: $$ \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25) $$ $$ 2x + 2y \frac{dy}{dx} = 0 $$
- Solve for $ \frac{dy}{dx} $ : $$ 2y \frac{dy}{dx} = -2x $$ $$ \frac{dy}{dx} = \frac{-2x}{2y} = \boxed{-\frac{x}{y}} $$
Chain Rule
The chain rule is essential for implicit differentiation. It states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.
In the context of implicit differentiation, the inner function is often $ y $ , which is a function of $ x $ .
For example, if we have a term like $ y^3 $ , we differentiate it as follows:
$$ \frac{d}{dx}(y^3) = 3y^2 \cdot \frac{dy}{dx} $$
Here, $ y^3 $ is the outer function, $ y $ is the inner function, and $ \frac{dy}{dx} $ represents the derivative of the inner function.
Applications of Implicit Differentiation
Implicit differentiation has various applications, including:
- Finding the slope of a tangent line to a curve defined implicitly.
- Determining the critical points of a function defined implicitly.
- Solving related rates problems involving implicit equations.
Summary
Implicit differentiation is a powerful tool for finding derivatives when a function is defined implicitly. By treating $ y $ as a function of $ x $ and applying the chain rule, we can successfully differentiate both sides of the equation and solve for $ \frac{dy}{dx} $ . This technique is valuable in various applications across calculus and other fields.