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Derivative of 3*e^x/log(x)
Identical expressions
three *e^x/log(x)
3 multiply by e to the power of x divide by logarithm of (x)
three multiply by e to the power of x divide by logarithm of (x)
3*ex/log(x)
3*ex/logx
3e^x/log(x)
3ex/log(x)
3ex/logx
3e^x/logx
3*e^x divide by log(x)

The derivative of the function/ 3*e^x/log(x)

Derivative of 3*e^x/log(x)

The solution

You have entered [src]

    x 
 3*E  
------
log(x)

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

(3*E^x)/log(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 3 e^{x}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of $e^{x}$ is itself.
  So, the result is: $3 e^{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

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

$\frac{3 \left(x \log{\left(x \right)} - 1\right) e^{x}}{x \log{\left(x \right)}^{2}}$

The answer is:

$\frac{3 \left(x \log{\left(x \right)} - 1\right) e^{x}}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    x          x  
 3*e        3*e   
------ - ---------
log(x)        2   
         x*log (x)

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

Simplify

The second derivative [src]

  /                     2   \   
  |               1 + ------|   
  |       2           log(x)|  x
3*|1 - -------- + ----------|*e 
  |    x*log(x)    2        |   
  \               x *log(x) /   
--------------------------------
             log(x)

$$\frac{3 \left(1 - \frac{2}{x \log{\left(x \right)}} + \frac{1 + \frac{2}{\log{\left(x \right)}}}{x^{2} \log{\left(x \right)}}\right) e^{x}}{\log{\left(x \right)}}$$

Simplify

The third derivative [src]

  /                 /      3         3   \                 \   
  |               2*|1 + ------ + -------|     /      2   \|   
  |                 |    log(x)      2   |   3*|1 + ------||   
  |       3         \             log (x)/     \    log(x)/|  x
3*|1 - -------- - ------------------------ + --------------|*e 
  |    x*log(x)           3                     2          |   
  \                      x *log(x)             x *log(x)   /   
---------------------------------------------------------------
                             log(x)

$$\frac{3 \left(1 - \frac{3}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right)}{x^{2} \log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{x^{3} \log{\left(x \right)}}\right) e^{x}}{\log{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of 3*e^x/log(x)

The graph:

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The solution