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The derivative of the function/ sinx*(logx/log7)

Derivative of sinx*(logx/log7)

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

The graph:

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The solution

You have entered [src] 
       log(x)
sin(x)*------
       log(7)
log⁡(x)log⁡(7)sin⁡(x)\frac{\log{\left(x \right)}}{\log{\left(7 \right)}} \sin{\left(x \right)} 
sin(x)*(log(x)/log(7))
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=log⁡(x)sin⁡(x)f{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} and g(x)=log⁡(7)g{\left(x \right)} = \log{\left(7 \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

      f(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

      g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      The result is: log⁡(x)cos⁡(x)+sin⁡(x)x\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of the constant log⁡(7)\log{\left(7 \right)} is zero.

    Now plug in to the quotient rule:

    log⁡(x)cos⁡(x)+sin⁡(x)xlog⁡(7)\frac{\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}}{\log{\left(7 \right)}}

  2. Now simplify:

    xlog⁡(x)cos⁡(x)+sin⁡(x)xlog⁡(7)\frac{x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \log{\left(7 \right)}}

The answer is:

xlog⁡(x)cos⁡(x)+sin⁡(x)xlog⁡(7)\frac{x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \log{\left(7 \right)}}

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 sin(x)    cos(x)*log(x)
-------- + -------------
x*log(7)       log(7)
log⁡(x)cos⁡(x)log⁡(7)+sin⁡(x)xlog⁡(7)\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\log{\left(7 \right)}} + \frac{\sin{\left(x \right)}}{x \log{\left(7 \right)}}
Simplify
The second derivative [src] 
  sin(x)                   2*cos(x)
- ------ - log(x)*sin(x) + --------
     2                        x    
    x                              
-----------------------------------
               log(7)
−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2log⁡(7)\frac{- \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}}{\log{\left(7 \right)}}
Simplify
The third derivative [src] 
                 3*sin(x)   3*cos(x)   2*sin(x)
-cos(x)*log(x) - -------- - -------- + --------
                    x           2          3   
                               x          x    
-----------------------------------------------
                     log(7)
−log⁡(x)cos⁡(x)−3sin⁡(x)x−3cos⁡(x)x2+2sin⁡(x)x3log⁡(7)\frac{- \log{\left(x \right)} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{x} - \frac{3 \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3}}}{\log{\left(7 \right)}}
Simplify
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