Question

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Please find the derivative of $\sf{\dfrac{e^{\frac{3}{x}}}{x^2}}$ . Step to step explanation!!​

Answer 1

Answer:

$\boxed{\sf{\dfrac{-e^{\dfrac{3}{x}} (3 + 2x )}{x^{4}}}}$

Find the derivative using the quotient rule:

$\sf{\dfrac{f(x)}{g(x)} = \dfrac{g(x) × f'(x) - f(x) × g'(x)}{(g(x))^{2}}}$

In this quotient

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

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

$\begin{gathered}e^{u} = u' ×e^{u}\\\\x^{n} = nx^{n-1}\end{gathered}$

quotient rule:

• $\sf{\dfrac{x^{2} × (e^{\dfrac{3}{x}} × (-3x^{-2})) - e^{\dfrac{3}{x}} × 2x }{(x^{2} )^{2}}}$

• $\sf{\dfrac{(e^{{{\dfrac{3}{x}}}} ×(-3)) - e^{\dfrac{3}{x}} × 2x }{(x^{2} )^{2}}}$

• $\sf{e^{\dfrac{3}{x}}}$
• $\dfrac{e^{\dfrac{3}{x}} (-3 - 2x )}{x^{4}}$

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