Question

What is the derivative of y=e^x/x?

Answer 1

`\displaystyle (dy)/(dx) = e^x \bigg( (1)/(x) - (1)/(x^2) \bigg)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]:
`\displaystyle (d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

Explanation:

Step 1: Define

Identify

`\displaystyle y = (e^x)/(x)`

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:
   `\displaystyle y' = ((e^x)'x - e^x(x)')/(x^2)`
2. Exponential Differentiation:
   `\displaystyle y' = (e^xx - e^x(x)')/(x^2)`
3. Basic Power Rule:
   `\displaystyle y' = (e^xx - e^x)/(x^2)`
4. Factor:
   `\displaystyle y' = (e^x(x - 1))/(x^2)`
5. Rewrite:
   `\displaystyle y' = e^x \bigg( (x)/(x^2) - (1)/(x^2) \bigg)`
6. Simplify:
   `\displaystyle y' = e^x \bigg( (1)/(x) - (1)/(x^2) \bigg)`

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

Unit: Differentiation