Answer:
[tex]\displaystyle \frac{dy}{dx} = \frac{2e^{2x} - 7}{5}[/tex]
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]: [tex]\displaystyle (b^m)^n = b^{m \cdot n}[/tex]
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: [tex]\displaystyle \frac{d}{dx}[e^u] = u'e^u[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle (e^x)^2 = 7x + 5y + 3[/tex]
Step 2: Differentiate
- Rewrite [Exponential Rule - Powering]: [tex]\displaystyle e^{2x} = 7x + 5y + 3[/tex]
- Implicit Differentiation: [tex]\displaystyle \frac{d}{dx}[e^{2x}] = \frac{d}{dx}[7x + 5y + 3][/tex]
- [Derivative] eˣ derivative: [tex]\displaystyle 2e^{2x} = \frac{d}{dx}[7x + 5y + 3][/tex]
- [Derivative] Basic Power Rule: [tex]\displaystyle 2e^{2x} = 1 \cdot 7x^{1 - 1} + 1 \cdot 5y^{1 - 1}\frac{dy}{dx}[/tex]
- [Derivative] Simplify: [tex]\displaystyle 2e^{2x} = 7 + 5\frac{dy}{dx}[/tex]
- [Derivative] [Subtraction Property of Equality] Subtract 7 on both sides: [tex]\displaystyle 2e^{2x} - 7 = 5\frac{dy}{dx}[/tex]
- [Derivative] [Division Property of Equality] Divide 5 on both sides: [tex]\displaystyle \frac{2e^{2x} - 7}{5} = \frac{dy}{dx}[/tex]
- [Derivative] Rewrite: [tex]\displaystyle \frac{dy}{dx} = \frac{2e^{2x} - 7}{5}[/tex]
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Derivatives - Implicit Differentiation
Book: College Calculus 10e
