Question

Please find the derivative of
`\displaystyle \frac{e^{(3)/(x)}}{x^2}`. Show all work necessary - thanks!

Answer 1

`\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)}}{x^4} - \frac{2e^{(3)/(x)}}{x^3}`

General Formulas and Concepts:

Pre-Algebra

• Splitting Fractions

Algebra I

• Terms/Coefficients
• Factoring
• Exponential Rule [Multiplying]:
  `\displaystyle b^m \cdot b^n = b^(m + n)`

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule:
`\displaystyle (d)/(dx) [e^u]=e^u \cdot u'`

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

Explanation:

Step 1: Define

`\displaystyle \frac{e^{(3)/(x)}}{x^2}\\f(x) = e^{(3)/(x)}\\g(x) = x^2`

Step 2: Differentiate

1. Quotient Rule:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{(d)/(dx)[e^{(3)/(x)}] \cdot x^2 - (d)/(dx)[x^2] \cdot e^{(3)/(x)}}{(x^2)^2}`
2. Derivative Rule:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{e^{(3)/(x)} \cdot (-3)/(x^2) \cdot x^2 - (d)/(dx)[x^2] \cdot e^{(3)/(x)}}{(x^2)^2}`
3. [Simplify] Multiply:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)} - (d)/(dx)[x^2] \cdot e^{(3)/(x)}}{(x^2)^2}`
4. Basic Power Rule:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)} - 2x^(2-1) \cdot e^{(3)/(x)}}{(x^2)^2}`
5. [Simplify] Subtract Exponents:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)} - 2x \cdot e^{(3)/(x)}}{(x^2)^2}`
6. [Simplify] Multiply:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)} - 2xe^{(3)/(x)}}{(x^2)^2}`
7. [Simplify] Exponent Rule:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)} - 2xe^{(3)/(x)}}{x^(2 + 2)}`
8. [Simplify] Add Exponents:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)} - 2xe^{(3)/(x)}}{x^4}`
9. [Simplify] Fraction Split:
   `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)}}{x^4} - \frac{2xe^{(3)/(x)}}{x^4}`
10. [Simplify - 2nd Fraction] Cancel Like Terms:
    `\displaystyle (d)/(dx)[\frac{e^{(3)/(x)}}{x^2}] = \frac{-3e^{(3)/(x)}}{x^4} - \frac{2e^{(3)/(x)}}{x^3}`

And we have our final answer!

Answer 2

Hello! :)

`\large\boxed{\frac{-e^{(3)/(x)} (3 + 2x )}{x^(4)}}`

Find the derivative using the quotient rule:

`(f(x))/(g(x)) = (g(x) * f'(x) - f(x) * g'(x))/((g(x))^(2))`

In this instance:

`f(x) = e^{(3)/(x) }\\\\g(x) = x^(2)`

Use the following properties to find the derivative of f(x) and g(x):

`e^(u) = u' * e^(u)\\\\x^(n) = nx^(n-1)`

Use the quotient rule:

`\frac{x^(2) * (e^{(3)/(x)} * (-3x^(-2))) - e^{(3)/(x)} * 2x }{(x^(2) )^(2)}`

Simplify the numerator:

`\frac{(e^{(3)/(x)} * (-3)) - e^{(3)/(x)} * 2x }{(x^(2) )^(2)}`

Factor out
`e^{(3)/(x)}`

`\frac{e^{(3)/(x)} (-3 - 2x )}{x^(4)}`

Factor out -1 from the numerator:

`\frac{-e^{(3)/(x)} (3 + 2x )}{x^(4)}`

And we're done! Thanks for posting the question to my 1000th answer!