Question

What is the derivative of e^(2x)

Answer 1

`\displaystyle (d)/(dx)[e^\big{2x}] = 2e^\big{2x}`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = e^\big{2x}`

Step 2: Differentiate

1. Exponential Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle (d)/(dx) = e^\big{2x} \cdot (d)/(dx)[2x]`
2. Basic Power Rule [Derivative Property - Multiplied Constant]:
   `\displaystyle (d)/(dx) = e^\big{2x} \cdot 2`
3. Simplify:
   `\displaystyle (d)/(dx) = 2e^\big{2x}`

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

Unit: Differentiation