Implicit Differentiation

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

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◇Overview

We say an equation determines $y$ as one or more implicit functions of $x$ when $y$ is not isolated on one side of the equation. For such implicit equations, we use a technique called implicit differentiation to solve for $\frac{dy}{dx}$ (or $y'$). This involves differentiating both sides of the equation with respect to $x$ and applying the chain rule whenever $y$ is differentiated.

◇Important Formulas

The Chain Rule in Implicit Differentiation

When differentiating a term involving $y$ with respect to $x$, apply the chain rule:
$\frac{d}{dx}[f(y)] = f'(y) \cdot \frac{dy}{dx}$

◇Worked Examples

General Procedure for Implicit Differentiation:

1. Differentiate both sides of the equation with respect to $x$.
2. Collect all terms containing $\frac{dy}{dx}$ on one side of the equation.
3. Factor out $\frac{dy}{dx}$ from these terms.
4. Solve for $\frac{dy}{dx}$ by dividing both sides by the factored expression.

Note: You can check your results by comparing them with explicit forms if it is possible to isolate $y$ and differentiate explicitly.

◇References & Navigation

• Previous Topic: ◇Higher Order Derivatives and Derivative of Trigonometric Functions
• Next Topic: ◇Derivative as Rate of Change and Related Rates Part 1
• Related Notes: ◇Differentiation Rules, ◇Linear Approximations and Differentials