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Derivative of:
Derivative of asin(t)
Derivative of -2*y
Derivative of log(sin(2*x+5))
Derivative of e^(-x)*log(x)
Limit of the function:
e^(-x)*log(x)
Identical expressions
e^(-x)*log(x)
e to the power of ( minus x) multiply by logarithm of (x)
e(-x)*log(x)
e-x*logx
e^(-x)log(x)
e(-x)log(x)
e-xlogx
e^-xlogx
Similar expressions
e^(x)*log(x)

The derivative of the function/ e^(-x)*log(x)

Derivative of e^(-x)*log(x)

The solution

You have entered [src]

 -x       
E  *log(x)

$$e^{- x} \log{\left(x \right)}$$

E^(-x)*log(x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = e^{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
Now plug in to the quotient rule:

$\left(- e^{x} \log{\left(x \right)} + \frac{e^{x}}{x}\right) e^{- 2 x}$
Now simplify:

$\frac{\left(- x \log{\left(x \right)} + 1\right) e^{- x}}{x}$

The answer is:

$\frac{\left(- x \log{\left(x \right)} + 1\right) e^{- x}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 -x             
e      -x       
--- - e  *log(x)
 x

$$- e^{- x} \log{\left(x \right)} + \frac{e^{- x}}{x}$$

Simplify

The second derivative [src]

/  1    2         \  -x
|- -- - - + log(x)|*e  
|   2   x         |    
\  x              /

$$\left(\log{\left(x \right)} - \frac{2}{x} - \frac{1}{x^{2}}\right) e^{- x}$$

Simplify

The third derivative [src]

/          2    3   3 \  -x
|-log(x) + -- + - + --|*e  
|           3   x    2|    
\          x        x /

$$\left(- \log{\left(x \right)} + \frac{3}{x} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) e^{- x}$$

Simplify

The graph

Derivative of e^(-x)*log(x)