Derivative of log(x*sin(x)+cos(x))

The solution

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

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

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

Detail solution

Let $u = x \sin{\left(x \right)} + \cos{\left(x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)$ :
1. Differentiate $x \sin{\left(x \right)} + \cos{\left(x \right)}$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $x \cos{\left(x \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of log(x*sin(x)+cos(x))

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The solution