Implicit Differentiation: 3.8

Implicit Differentiation

Introduction to Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not given explicitly as y = f(x), but rather as a relationship between x and y. This method is essential when dealing with equations where y cannot be easily isolated.

• Explicit Function: A function where y is directly expressed in terms of x, e.g., $y = f(x)$.

• Implicit Function: A function where y and x are related by an equation, e.g., $F(x, y) = 0$.

• Implicit differentiation treats y as a function of x and applies the chain rule when differentiating terms involving y.

Basic Steps in Implicit Differentiation

1. Differentiate both sides of the equation with respect to x, treating y as a function of x.

2. Apply the chain rule to terms involving y: $rac{d}{dx}[y] = y'$.

3. Solve for $y'$ (the derivative of y with respect to x).

Worked Examples

Example 1: Circle Equation

Given $x^2 + y^2 = 25$, find $y'$.

• Differentiate both sides: $(x^2 + y^2)' = 0$

• Apply the chain rule: $2x + 2y y' = 0$

• Solve for $y'$: $2y y' = -2x$

• $y' = -\frac{2x}{2y} = -\frac{x}{y}$

Example 2: Rational Function

Given $y = \frac{x + 3}{y - 3}$, find $y'$.

• Multiply both sides by $(y - 3)$: $y(y - 3) = x + 3$

• Expand: $y^2 - 3y = x + 3$

• Differentiate: $2y y' - 3 y' = 1$

• Factor $y'$: $y'(2y - 3) = 1$

• $y' = \frac{1}{2y - 3}$

Example 3: Trigonometric Functions

Given $\sin x + \sin y = y$, find $y'$.

• Differentiate both sides: $\cos x + \cos y y' = y'$

• Rearrange: $\cos x = y' - \cos y y'$

• Factor $y'$: $\cos x = y'(1 - \cos y)$

• $y' = \frac{\cos x}{1 - \cos y}$

Example 4: Exponential Functions

Given $e^{xy} = 3y$, find $y'$.

• Differentiate both sides: $e^{xy}(x y' + y) = 3 y'$

• Expand: $y e^{xy} + x e^{xy} y' = 3 y'$

• Group $y'$ terms: $x e^{xy} y' - 3 y' = -y e^{xy}$

• Factor $y'$: $y' (x e^{xy} - 3) = -y e^{xy}$

• $y' = \frac{y e^{xy}}{3 - x e^{xy}}$

Example 5: Product and Trigonometric Functions

Given $y = e^x \tan y$, find $y'$.

• Differentiate both sides: $y' = e^x \tan y + e^x \sec^2 y y'$

• Group $y'$ terms: $y' - e^x \sec^2 y y' = e^x \tan y$

• Factor $y'$: $y'(1 - e^x \sec^2 y) = e^x \tan y$

• $y' = \frac{e^x \tan y}{1 - e^x \sec^2 y}$

Example 6: Mixed Trigonometric and Exponential Functions

Given $\cos y^2 + x = e^y$, find $y'$.

• Differentiate both sides: $-\sin y^2 \cdot 2y y' + 1 = e^y y'$

• Expand: $-2y \sin y^2 y' + 1 = e^y y'$

• Group $y'$ terms: $1 = e^y y' + 2y \sin y^2 y'$

• Factor $y'$: $1 = y'(e^y + 2y \sin y^2)$

• $y' = \frac{1}{e^y + 2y \sin y^2}$

Example 7: Slope of a Curve at a Point

Find the slope of the curve $4x^2 + xy + 5y^2 = 52$ at the point $(1, 3)$.

• Differentiate both sides: $8x + y + x y' + 10y y' = 0$

• Group $y'$ terms: $x y' + 10y y' = -8x - y$

• Factor $y'$: $(x + 10y) y' = -8x - y$

• $y' = \frac{-8x - y}{x + 10y}$

• At $(1, 3)$: $y' = \frac{-8 \cdot 1 - 3}{1 + 10 \cdot 3} = \frac{-8 - 3}{1 + 30} = \frac{-11}{31}$

Key Formulas and Rules Used in Implicit Differentiation

• Chain Rule: $\frac{d}{dx}[f(y)] = f'(y) y'$

• Product Rule: $\frac{d}{dx}[u v] = u' v + u v'$

• Quotient Rule: $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u' v - u v'}{v^2}$

• Trigonometric Derivatives:

  • $\frac{d}{dx}[\sin u] = \cos u \cdot u'$

  • $\frac{d}{dx}[\tan u] = \sec^2 u \cdot u'$

  • $\frac{d}{dx}[\cos u] = -\sin u \cdot u'$

• Exponential Derivatives: $\frac{d}{dx}[e^{u}] = e^{u} \cdot u'$

Summary Table: Common Implicit Differentiation Results

Equation	Derivative $y'$
$x^2 + y^2 = 25$	$-\frac{x}{y}$
$y = \frac{x + 3}{y - 3}$	$\frac{1}{2y - 3}$
$\sin x + \sin y = y$	$\frac{\cos x}{1 - \cos y}$
$e^{xy} = 3y$	$\frac{y e^{xy}}{3 - x e^{xy}}$
$y = e^x \tan y$	$\frac{e^x \tan y}{1 - e^x \sec^2 y}$
$\cos y^2 + x = e^y$	$\frac{1}{e^y + 2y \sin y^2}$
$4x^2 + xy + 5y^2 = 52$ at $(1,3)$	$-\frac{11}{31}$

Applications of Implicit Differentiation

• Finding slopes of curves defined by implicit equations.

• Analyzing relationships where variables cannot be easily separated.

• Solving problems in physics, engineering, and economics where constraints are given implicitly.

Additional info: The notes above expand on the brief slide content, providing full explanations, step-by-step solutions, and context for each example. The summary table is constructed to aid quick review and comparison of results.