Derivative of sin(x)*loga(x)

The solution

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       log(x)
sin(x)*------
       log(a)

$$\frac{\log{\left(x \right)}}{\log{\left(a \right)}} \sin{\left(x \right)}$$

sin(x)*(log(x)/log(a))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(a \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(a \right)}$ is zero.
Now plug in to the quotient rule:

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

$\frac{x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \log{\left(a \right)}}$

The answer is:

$\frac{x \log{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{x \log{\left(a \right)}}$

The first derivative [src]

 sin(x)    cos(x)*log(x)
-------- + -------------
x*log(a)       log(a)

$$\frac{\log{\left(x \right)} \cos{\left(x \right)}}{\log{\left(a \right)}} + \frac{\sin{\left(x \right)}}{x \log{\left(a \right)}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                 3*sin(x)   3*cos(x)   2*sin(x)
-cos(x)*log(x) - -------- - -------- + --------
                    x           2          3   
                               x          x    
-----------------------------------------------
                     log(a)

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

Simplify

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Derivative of sin(x)*loga(x)

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