Derivative of y=xsinxlnx

The solution

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x*sin(x)*log(x)

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

(x*sin(x))*log(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; 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: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
The result is: $\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}$

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=xsinxlnx

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Derivative of y=x*sinx*lnx

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Piecewise:

The solution

Derivative of y=xsinxlnx