Question

differentiation. solve ​

Answer 1

Step-by-step explanation:

Given,

$\rm f(x) = \left( \dfrac{ {e}^{x} }{x} \right)$

Using the quotient rule, according to which:

$\boxed { \tt \dfrac{d}{dx} \left(\dfrac{f(x)}{g(x)} \right) = \dfrac{g(x)f'(x) - g'(x)f(x)}{ {g}^{2}(x) } }$

Therefore,

$\rm f'(x) = \left( \dfrac{x \cdot \frac{d}{dx}({e}^{x}) - \frac{d}{dx}( x) {e}^{x} }{ {x}^{2} } \right)$

We know that, d/dx (eˣ) = eˣ and d/dx (x) = 1.

$\rm f'(x) = \left( \dfrac{x{e}^{x}- {e}^{x} }{ {x}^{2} } \right)$

$\rm f'(x) = \dfrac{{e}^{x}(x - 1)}{ {x}^{2} }$

This is the required answer.

Additional Information:

Some standard differentiation formulas,

• d/dx (x) = 1
• d/dx (constant) = 0
• d/dx (log x) = 1/x
• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec²x
• d/dx (cosec x) = -cosec(x) cot(x)
• d/dx (sec x) = sec(x) tan(x)
• d/dx (cot x) = -cosec²(x)
• d/dx (eˣ) = eˣ
• d/dx ( f(g(x)) = f'(g(x)) × g'(x)