Answer:
[tex]\displaystyle \frac{dy}{dx} = 6e^\big{5x}(5x + 1)[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = 6xe^\big{5x}[/tex]
Step 2: Differentiate
- Derivative Rule [Product Rule]: [tex]\displaystyle y' = \frac{d}{dx}[6x]e^\big{5x} + 6x\frac{d}{dx}[e^\big{5x}][/tex]
- Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle y' = 6\frac{d}{dx}[x]e^\big{5x} + 6x\frac{d}{dx}[e^\big{5x}][/tex]
- Basic Power Rule: [tex]\displaystyle y' = 6e^\big{5x} + 6x\frac{d}{dx}[e^\big{5x}][/tex]
- Exponential Differentiation [Derivative Rule - Chain Rule]: [tex]\displaystyle y' = 6e^\big{5x} + 6xe^\big{5x}\frac{d}{dx}[5x][/tex]
- Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle y' = 6e^\big{5x} + 30xe^\big{5x}\frac{d}{dx}[x][/tex]
- Basic Power Rule: [tex]\displaystyle y' = 6e^\big{5x} + 30xe^\big{5x}[/tex]
- Factor: [tex]\displaystyle y' = 6e^\big{5x}(5x + 1)[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
