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Derivative of:
Derivative of sin(2*x^2+3)
Derivative of (1+x)/(4-x^2)
Derivative of -1/(x-1)^2
Derivative of (1+tan(3*x)^(2))*e^(-x/2)
Identical expressions
log(five *x)*e^(-x)
logarithm of (5 multiply by x) multiply by e to the power of ( minus x)
logarithm of (five multiply by x) multiply by e to the power of ( minus x)
log(5*x)*e(-x)
log5*x*e-x
log(5x)e^(-x)
log(5x)e(-x)
log5xe-x
log5xe^-x
Similar expressions
log(5*x)*e^(x)

The derivative of the function/ log(5*x)*e^(-x)

Derivative of log(5x)e^(-x)

The solution

You have entered [src]

          -x
log(5*x)*E

$$e^{- x} \log{\left(5 x \right)}$$

log(5*x)*E^(-x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(5 x \right)}$ and $g{\left(x \right)} = e^{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 5 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $5$
  The result of the chain rule 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:

$\left(- e^{x} \log{\left(5 x \right)} + \frac{e^{x}}{x}\right) e^{- 2 x}$
Now simplify:

$\frac{\left(- x \log{\left(5 x \right)} + 1\right) e^{- x}}{x}$

The answer is:

$\frac{\left(- x \log{\left(5 x \right)} + 1\right) e^{- x}}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 -x               
e      -x         
--- - e  *log(5*x)
 x

$$- e^{- x} \log{\left(5 x \right)} + \frac{e^{- x}}{x}$$

Simplify

The second derivative [src]

/  1    2           \  -x
|- -- - - + log(5*x)|*e  
|   2   x           |    
\  x                /

$$\left(\log{\left(5 x \right)} - \frac{2}{x} - \frac{1}{x^{2}}\right) e^{- x}$$

Simplify

The third derivative [src]

/            2    3   3 \  -x
|-log(5*x) + -- + - + --|*e  
|             3   x    2|    
\            x        x /

$$\left(- \log{\left(5 x \right)} + \frac{3}{x} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) e^{- x}$$

Simplify