Question

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

Answer 1

Hello! :)

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

Find the derivative using the quotient rule:

$\frac{f(x)}{g(x)} = \frac{g(x) * f'(x) - f(x) * g'(x)}{(g(x))^{2}}$

In this instance:

$f(x) = e^{\frac{3}{x} }\\\\g(x) = x^{2}$

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:

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

Simplify the numerator:

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

Factor out $e^{\frac{3}{x}}$

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

Factor out -1 from the numerator:

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

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

Answer 2

Answer:

$\displaystyle \frac{d}{dx}[\frac{e^{\frac{3}{x}}}{x^2}] = \frac{-3e^{\frac{3}{x}}}{x^4} - \frac{2e^{\frac{3}{x}}}{x^3}$

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:    $\displaystyle \frac{d}{dx} [e^u]=e^u \cdot u'$

Quotient Rule:      $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

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

Step 2: Differentiate

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

And we have our final answer!