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The derivative of the function/ 1/2*log(x*sin(x)+cos(x))^2

Derivative of 1/2log(xsin(x)+cos(x))^2

The solution

You have entered [src]

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

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

log(x*sin(x) + cos(x))^2/2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)}$ :
  1. Let $u = x \sin{\left(x \right)} + \cos{\left(x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. 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 $\frac{d}{d x} f{\left(x \right)}$ :
        
        Apply the power rule: $x$ goes to $1$
        
        $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
        
        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 result of the chain rule is:
  
  $\frac{2 x \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$
So, the result is: $\frac{x \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)} \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

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Derivative of 1/2log(xsin(x)+cos(x))^2

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Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of 1/2*log(x*sin(x)+cos(x))^2

The graph:

Piecewise:

The solution

Derivative of 1/2log(xsin(x)+cos(x))^2