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

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

The solution

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                                              2    
                           ___________________     
log(x*sin(x) + cos(x)) + \/ x*sin(x) + cos(x)   + 1

$$\left(\left(\sqrt{x \sin{\left(x \right)} + \cos{\left(x \right)}}\right)^{2} + \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)}\right) + 1$$

log(x*sin(x) + cos(x)) + (sqrt(x*sin(x) + cos(x)))^2 + 1

Detail solution

Differentiate $\left(\left(\sqrt{x \sin{\left(x \right)} + \cos{\left(x \right)}}\right)^{2} + \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)}\right) + 1$ term by term:
1. Differentiate $\left(\sqrt{x \sin{\left(x \right)} + \cos{\left(x \right)}}\right)^{2} + \log{\left(x \sin{\left(x \right)} + \cos{\left(x \right)} \right)}$ term by term:
  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:
    
    $\frac{x \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$
  4. Let $u = \sqrt{x \sin{\left(x \right)} + \cos{\left(x \right)}}$ .
  5. Apply the power rule: $u^{2}$ goes to $2 u$
  6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x \sin{\left(x \right)} + \cos{\left(x \right)}}$ :
    1. Let $u = x \sin{\left(x \right)} + \cos{\left(x \right)}$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    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:
        
        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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
        
        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:
      
      $\frac{x \cos{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + \cos{\left(x \right)}}}$
    The result of the chain rule is:
    
    $x \cos{\left(x \right)}$
  The result is: $x \cos{\left(x \right)} + \frac{x \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$
2. The derivative of the constant $1$ is zero.
The result is: $x \cos{\left(x \right)} + \frac{x \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$
Now simplify:

$\frac{x \left(x \sin{\left(x \right)} + \cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$

The answer is:

$\frac{x \left(x \sin{\left(x \right)} + \cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

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