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Derivative of e^(3*x)/x^2
Limit of the function:
e^(3*x)/x^2
Identical expressions
e^(three *x)/x^ two
e to the power of (3 multiply by x) divide by x squared
e to the power of (three multiply by x) divide by x to the power of two
e(3*x)/x2
e3*x/x2
e^(3*x)/x²
e to the power of (3*x)/x to the power of 2
e^(3x)/x^2
e(3x)/x2
e3x/x2
e^3x/x^2
e^(3*x) divide by x^2

The derivative of the function/ e^(3*x)/x^2

Derivative of e^(3*x)/x^2

The solution

You have entered [src]

 3*x
E   
----
  2 
 x

$$\frac{e^{3 x}}{x^{2}}$$

E^(3*x)/x^2

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = e^{3 x}$ and $g{\left(x \right)} = x^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
  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: $3$
  The result of the chain rule is:
  
  $3 e^{3 x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
Now plug in to the quotient rule:

$\frac{3 x^{2} e^{3 x} - 2 x e^{3 x}}{x^{4}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3*x      3*x
  2*e      3*e   
- ------ + ------
     3        2  
    x        x

$$\frac{3 e^{3 x}}{x^{2}} - \frac{2 e^{3 x}}{x^{3}}$$

Simplify

The second derivative [src]

  /    4   2 \  3*x
3*|3 - - + --|*e   
  |    x    2|     
  \        x /     
-------------------
          2        
         x

$$\frac{3 \left(3 - \frac{4}{x} + \frac{2}{x^{2}}\right) e^{3 x}}{x^{2}}$$

Simplify

The third derivative [src]

  /    18   8    18\  3*x
3*|9 - -- - -- + --|*e   
  |    x     3    2|     
  \         x    x /     
-------------------------
             2           
            x

$$\frac{3 \left(9 - \frac{18}{x} + \frac{18}{x^{2}} - \frac{8}{x^{3}}\right) e^{3 x}}{x^{2}}$$

Simplify

The graph

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Identical expressions

Derivative of e^(3*x)/x^2

The graph:

Piecewise:

The solution