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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let .

      2. Then, apply the chain rule. Multiply by :

        1. The derivative of cosine is negative sine:

        The result of the chain rule is:

      So, the result is:

    To find :

    1. The derivative of is .

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
  cos(x)     cos(x)              
 7          7      *log(7)*sin(x)
--------- + ---------------------
     2              log(x)       
x*log (x)                        
$$\frac{7^{\cos{\left(x \right)}} \log{\left(7 \right)} \sin{\left(x \right)}}{\log{\left(x \right)}} + \frac{7^{\cos{\left(x \right)}}}{x \log{\left(x \right)}^{2}}$$
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)                                   
$$- \frac{7^{\cos{\left(x \right)}} \left(\left(\log{\left(7 \right)} \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(7 \right)} + \frac{2 \log{\left(7 \right)} \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{1 + \frac{2}{\log{\left(x \right)}}}{x^{2} \log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$
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)                                                                             
$$\frac{7^{\cos{\left(x \right)}} \left(- \left(- \log{\left(7 \right)}^{2} \sin^{2}{\left(x \right)} + 3 \log{\left(7 \right)} \cos{\left(x \right)} + 1\right) \log{\left(7 \right)} \sin{\left(x \right)} + \frac{3 \left(\log{\left(7 \right)} \sin^{2}{\left(x \right)} - \cos{\left(x \right)}\right) \log{\left(7 \right)}}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \log{\left(7 \right)} \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{x^{3} \log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$