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Similar expressions
(2*x-x^2)/x^(3*x)

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

Derivative of (2x+x^2)/x^(3x)

The solution

You have entered [src]

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

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

(2*x + x^2)/x^(3*x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + 2 x$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  2. 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: $2$
  The result is: $2 x + 2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Don't know the steps in finding this derivative.
  
  But the derivative is
  
  $\left(3 x\right)^{3 x} \left(\log{\left(3 x \right)} + 1\right)$
Now plug in to the quotient rule:

$x^{- 6 x} \left(x^{3 x} \left(2 x + 2\right) - \left(3 x\right)^{3 x} \left(x^{2} + 2 x\right) \left(\log{\left(3 x \right)} + 1\right)\right)$
Now simplify:

$x^{- 6 x} \left(- x \left(3 x\right)^{3 x} \left(x + 2\right) \left(\log{\left(3 x \right)} + 1\right) + 2 x^{3 x} \left(x + 1\right)\right)$

The answer is:

$x^{- 6 x} \left(- x \left(3 x\right)^{3 x} \left(x + 2\right) \left(\log{\left(3 x \right)} + 1\right) + 2 x^{3 x} \left(x + 1\right)\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

 -3*x /                                          /  1                 2\\
x    *|2 - 12*(1 + x)*(1 + log(x)) + 3*x*(2 + x)*|- - + 3*(1 + log(x)) ||
      \                                          \  x                  //

$$x^{- 3 x} \left(3 x \left(x + 2\right) \left(3 \left(\log{\left(x \right)} + 1\right)^{2} - \frac{1}{x}\right) - 12 \left(x + 1\right) \left(\log{\left(x \right)} + 1\right) + 2\right)$$

Simplify

The third derivative [src]

   -3*x /                          /  1                 2\             /1                  3   9*(1 + log(x))\\
3*x    *|-6 - 6*log(x) + 6*(1 + x)*|- - + 3*(1 + log(x)) | + x*(2 + x)*|-- - 9*(1 + log(x))  + --------------||
        |                          \  x                  /             | 2                           x       ||
        \                                                              \x                                    //

$$3 x^{- 3 x} \left(x \left(x + 2\right) \left(- 9 \left(\log{\left(x \right)} + 1\right)^{3} + \frac{9 \left(\log{\left(x \right)} + 1\right)}{x} + \frac{1}{x^{2}}\right) + 6 \left(x + 1\right) \left(3 \left(\log{\left(x \right)} + 1\right)^{2} - \frac{1}{x}\right) - 6 \log{\left(x \right)} - 6\right)$$

Simplify

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Derivative of (2x+x^2)/x^(3x)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

Derivative of (2x+x^2)/x^(3x)