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(4ln^ two (8x+ seven))/exp^x
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Similar expressions
(4ln^2(8x-7))/exp^x

The derivative of the function/ (4ln^2(8x+7))/exp^x

Derivative of (4ln^2(8x+7))/exp^x

The solution

You have entered [src]

     2         
4*log (8*x + 7)
---------------
        x      
       E

$$\frac{4 \log{\left(8 x + 7 \right)}^{2}}{e^{x}}$$

(4*log(8*x + 7)^2)/E^x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 4 \log{\left(8 x + 7 \right)}^{2}$ and $g{\left(x \right)} = e^{x}$ .

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. Let $u = \log{\left(8 x + 7 \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(8 x + 7 \right)}$ :
    1. Let $u = 8 x + 7$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(8 x + 7\right)$ :
      1. Differentiate $8 x + 7$ term by term:
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $8$
        
        The derivative of the constant $7$ is zero.
        
        The result is: $8$
      The result of the chain rule is:
      
      $\frac{8}{8 x + 7}$
    The result of the chain rule is:
    
    $\frac{16 \log{\left(8 x + 7 \right)}}{8 x + 7}$
  So, the result is: $\frac{64 \log{\left(8 x + 7 \right)}}{8 x + 7}$
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:

$\left(- 4 e^{x} \log{\left(8 x + 7 \right)}^{2} + \frac{64 e^{x} \log{\left(8 x + 7 \right)}}{8 x + 7}\right) e^{- 2 x}$
Now simplify:

$\frac{4 \left(- \left(8 x + 7\right) \log{\left(8 x + 7 \right)} + 16\right) e^{- x} \log{\left(8 x + 7 \right)}}{8 x + 7}$

The answer is:

$\frac{4 \left(- \left(8 x + 7\right) \log{\left(8 x + 7 \right)} + 16\right) e^{- x} \log{\left(8 x + 7 \right)}}{8 x + 7}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                            -x             
       2           -x   64*e  *log(8*x + 7)
- 4*log (8*x + 7)*e   + -------------------
                              8*x + 7

$$- 4 e^{- x} \log{\left(8 x + 7 \right)}^{2} + \frac{64 e^{- x} \log{\left(8 x + 7 \right)}}{8 x + 7}$$

Simplify

The second derivative [src]

  /   2            128*(-1 + log(7 + 8*x))   32*log(7 + 8*x)\  -x
4*|log (7 + 8*x) - ----------------------- - ---------------|*e  
  |                                2             7 + 8*x    |    
  \                       (7 + 8*x)                         /

$$4 \left(\log{\left(8 x + 7 \right)}^{2} - \frac{32 \log{\left(8 x + 7 \right)}}{8 x + 7} - \frac{128 \left(\log{\left(8 x + 7 \right)} - 1\right)}{\left(8 x + 7\right)^{2}}\right) e^{- x}$$

Simplify

The third derivative [src]

  /     2            48*log(7 + 8*x)   384*(-1 + log(7 + 8*x))   1024*(-3 + 2*log(7 + 8*x))\  -x
4*|- log (7 + 8*x) + --------------- + ----------------------- + --------------------------|*e  
  |                      7 + 8*x                       2                          3        |    
  \                                           (7 + 8*x)                  (7 + 8*x)         /

$$4 \left(- \log{\left(8 x + 7 \right)}^{2} + \frac{48 \log{\left(8 x + 7 \right)}}{8 x + 7} + \frac{384 \left(\log{\left(8 x + 7 \right)} - 1\right)}{\left(8 x + 7\right)^{2}} + \frac{1024 \left(2 \log{\left(8 x + 7 \right)} - 3\right)}{\left(8 x + 7\right)^{3}}\right) e^{- x}$$

Simplify

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Derivative of (4ln^2(8x+7))/exp^x

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The solution