Question

Find dy/dx by implicit differentiation for the following equation. e^x^2=7x+5y+3

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} = \frac{2e^{2x} - 7}{5}$

General Formulas and Concepts:

Pre-Algebra

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Algebra I

• Exponential Rule [Powering]: $\displaystyle (b^m)^n = b^{m \cdot n}$

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Implicit Differentiation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

eˣ Derivative: $\displaystyle \frac{d}{dx}[e^u] = u'e^u$

Step-by-step explanation:

Step 1: Define

$\displaystyle (e^x)^2 = 7x + 5y + 3$

Step 2: Differentiate

1. Rewrite [Exponential Rule - Powering]:                                                          $\displaystyle e^{2x} = 7x + 5y + 3$
2. Implicit Differentiation:                                                                                   $\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[7x + 5y + 3]$
3. [Derivative] eˣ derivative:                                                                                $\displaystyle 2e^{2x} = \frac{d}{dx}[7x + 5y + 3]$
4. [Derivative] Basic Power Rule:                                                                       $\displaystyle 2e^{2x} = 1 \cdot 7x^{1 - 1} + 1 \cdot 5y^{1 - 1}\frac{dy}{dx}$
5. [Derivative] Simplify:                                                                                       $\displaystyle 2e^{2x} = 7 + 5\frac{dy}{dx}$
6. [Derivative] [Subtraction Property of Equality] Subtract 7 on both sides:  $\displaystyle 2e^{2x} - 7 = 5\frac{dy}{dx}$
7. [Derivative] [Division Property of Equality] Divide 5 on both sides:            $\displaystyle \frac{2e^{2x} - 7}{5} = \frac{dy}{dx}$
8. [Derivative] Rewrite:                                                                                       $\displaystyle \frac{dy}{dx} = \frac{2e^{2x} - 7}{5}$

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Derivatives - Implicit Differentiation

Book: College Calculus 10e