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Implicit Differentiation Practice Problems

Implicit Differentiation: A Comprehensive Guide for Beginners

Introduction

Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly in terms of another variable. This method is especially useful when it is difficult or impossible to solve the equation explicitly for one variable.

Method 1: Solving for y and Differentiating Directly

  1. Solve the equation for y in terms of x.
  2. Differentiate the resulting equation with respect to x.

Method 2: Implicit Differentiation

  1. Differentiate both sides of the equation with respect to x, treating y as a function of x.
  2. Solve the resulting equation for \frac{dy}{dx}.

Examples

Example 1

Find \frac{dy}{dx} for y2 + x2 = 4. Using Method 1:
  1. Solve for y: y = \pm\sqrt{4 - x^2}
  2. Differentiate: \frac{dy}{dx} = \frac{d}{dx}(\pm\sqrt{4 - x^2}) \frac{dy}{dx} = \pm\frac{-x}{\sqrt{4 - x^2}}
Using Method 2:
  1. Differentiate both sides: 2y\frac{dy}{dx} + 2x = 0
  2. Solve for \frac{dy}{dx}: \frac{dy}{dx} = \frac{-x}{y}

Example 2

Find \frac{d^2y}{dx^2} for y3 - x3 = 4. Using implicit differentiation:
  1. Differentiate both sides: 3y2\frac{dy}{dx} - 3x2 = 0
  2. Differentiate again: 6y\frac{dy}{dx}^2 + 3y2\frac{d^2y}{dx^2} - 6x = 0
  3. Solve for \frac{d^2y}{dx^2}: \frac{d^2y}{dx^2} = \frac{-2x}{3y2(1 + \frac{dy}{dx}^2)}


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