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 $\backslash frac\{dy\}\{dx\}$.

Examples

Example 1

Find $\backslash frac\{dy\}\{dx\}$ for y² + x² = 4. Using Method 1:

1. Solve for y: y = $\backslash pm$$\backslash sqrt\{4\; -\; x^2\}$
2. Differentiate: $\backslash frac\{dy\}\{dx\}$ = $\backslash frac\{d\}\{dx\}(\backslash pm\backslash sqrt\{4\; -\; x^2\})$ $\backslash frac\{dy\}\{dx\}$ = $\backslash pm\backslash frac\{-x\}\{\backslash sqrt\{4\; -\; x^2\}\}$

Using Method 2:

1. Differentiate both sides: 2y$\backslash frac\{dy\}\{dx\}$ + 2x = 0
2. Solve for $\backslash frac\{dy\}\{dx\}$: $\backslash frac\{dy\}\{dx\}$ = $\backslash frac\{-x\}\{y\}$

Example 2

Find $\backslash frac\{d^2y\}\{dx^2\}$ for y³ - x³ = 4. Using implicit differentiation:

1. Differentiate both sides: 3y²$\backslash frac\{dy\}\{dx\}$ - 3x² = 0
2. Differentiate again: 6y$\backslash frac\{dy\}\{dx\}^2$ + 3y²$\backslash frac\{d^2y\}\{dx^2\}$ - 6x = 0
3. Solve for $\backslash frac\{d^2y\}\{dx^2\}$: $\backslash frac\{d^2y\}\{dx^2\}$ = $\backslash frac\{-2$x}{3y²(1 + $\backslash frac\{dy\}\{dx\}^2$)}