Question

Diffentiate Log(e^x/e) with respective to x

Answer 1

Answer:

1

Step-by-step explanation:

Function we are given to differentiate is,

$\longrightarrow f(x) = \log _e \left(\dfrac{ {e}^{x} }{e} \right)$

Now, we will be using the following property of logarithm function.

• $\red{ \boxed{\log \left( \frac{a}{b} \right ) = \log(a) - \log(b)}}$
• $\red{\boxed{ \log_x(x) = 1}}$
• $\red{\boxed{\log( {y}^{x} ) =x \log(y)}}$

Therefore we get:

${ \longrightarrow f(x) = \log _e \left( {e}^{x} \right) - \log_e(e)}$

${ \longrightarrow f(x) =x \log _e \left( {e} \right) - 1}$

${ \longrightarrow f(x) =x (1) - 1}$

${ \longrightarrow f(x) =x - 1}$

Now differentiating both sides with respect to x.

${ \longrightarrow f'(x) = \dfrac{d}{dx}(x - 1)}$

${ \longrightarrow f'(x) = \dfrac{d}{dx}(x ) - \dfrac{d}{dx}(1)}$

Now, we will use formulas of derivative:

• $\blue{\boxed{ \frac{d}{dx}( {x}^{n}) = n {x}^{n - 1} }}$
• $\blue{ \boxed{ \frac{d}{dx}({ \sf{constant}}) =0 }}$

${ \longrightarrow f'(x) = 1 - 0}$

$\longrightarrow f'(x) = 1$

Therefore the required answer is:

$\underline{ \boxed{ \frac{d}{dx} \: \log_e \left( \frac{ {e}^{x} }{e} \right) = 1}}$