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Similar expressions
(7^cos(x))/ln(x)
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The derivative of the function/ (-7^cos(x))/ln(x)

Derivative of (-7^cos(x))/ln(x)

The solution

You have entered [src]

  cos(x) 
-7       
---------
  log(x)

$$\frac{\left(-1\right) 7^{\cos{\left(x \right)}}}{\log{\left(x \right)}}$$

(-7^cos(x))/log(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 7^{\cos{\left(x \right)}}$ 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. Let $u = \cos{\left(x \right)}$ .
  2. $\frac{d}{d u} 7^{u} = 7^{u} \log{\left(7 \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- 7^{\cos{\left(x \right)}} \log{\left(7 \right)} \sin{\left(x \right)}$
  So, the result is: $7^{\cos{\left(x \right)}} \log{\left(7 \right)} \sin{\left(x \right)}$
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:

$\frac{7^{\cos{\left(x \right)}} \log{\left(7 \right)} \log{\left(x \right)} \sin{\left(x \right)} + \frac{7^{\cos{\left(x \right)}}}{x}}{\log{\left(x \right)}^{2}}$
Now simplify:

$\frac{7^{\cos{\left(x \right)}} \left(x \log{\left(7 \right)} \log{\left(x \right)} \sin{\left(x \right)} + 1\right)}{x \log{\left(x \right)}^{2}}$

The answer is:

$\frac{7^{\cos{\left(x \right)}} \left(x \log{\left(7 \right)} \log{\left(x \right)} \sin{\left(x \right)} + 1\right)}{x \log{\left(x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  cos(x)     cos(x)              
 7          7      *log(7)*sin(x)
--------- + ---------------------
     2              log(x)       
x*log (x)

$$\frac{7^{\cos{\left(x \right)}} \log{\left(7 \right)} \sin{\left(x \right)}}{\log{\left(x \right)}} + \frac{7^{\cos{\left(x \right)}}}{x \log{\left(x \right)}^{2}}$$

Simplify

The second derivative [src]

         /                                          2                     \ 
         |                                    1 + ------                  | 
  cos(x) |/             2          \              log(x)   2*log(7)*sin(x)| 
-7      *|\-cos(x) + sin (x)*log(7)/*log(7) + ---------- + ---------------| 
         |                                     2               x*log(x)   | 
         \                                    x *log(x)                   / 
----------------------------------------------------------------------------
                                   log(x)

$$- \frac{7^{\cos{\left(x \right)}} \left(\left(\log{\left(7 \right)} \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(7 \right)} + \frac{2 \log{\left(7 \right)} \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{1 + \frac{2}{\log{\left(x \right)}}}{x^{2} \log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$

Simplify

The third derivative [src]

        /                                                            /      3         3   \                                                                     \
        |                                                          2*|1 + ------ + -------|                                           /      2   \              |
        |                                                            |    log(x)      2   |     /             2          \          3*|1 + ------|*log(7)*sin(x)|
 cos(x) |  /       2       2                     \                   \             log (x)/   3*\-cos(x) + sin (x)*log(7)/*log(7)     \    log(x)/              |
7      *|- \1 - log (7)*sin (x) + 3*cos(x)*log(7)/*log(7)*sin(x) + ------------------------ + ----------------------------------- + ----------------------------|
        |                                                                  3                                x*log(x)                          2                 |
        \                                                                 x *log(x)                                                          x *log(x)          /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                              log(x)

$$\frac{7^{\cos{\left(x \right)}} \left(- \left(- \log{\left(7 \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(7 \right)} \cos{\left(x \right)} + 1\right) \log{\left(7 \right)} \sin{\left(x \right)} + \frac{3 \left(\log{\left(7 \right)} \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(7 \right)}}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(7 \right)} \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{x^{3} \log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$

Simplify

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Derivative of (-7^cos(x))/ln(x)

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The solution