Question

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

Answer 1

Answer:

These are called second partial derivatives, and the notation is analogous to the d 2 f d x 2 \ dfrac{d^2 f}{dx^2} dx2d2fstart fraction

Answer 2

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

$f ^\prime(x) = \frac{d}{dx} ( \frac{ {e}^{ \frac{3}{x} } }{ {x}^{2} } )$

Using the Quotient Rule of Differentiation

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

Using The Chain Rule of Differentiation

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

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

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

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