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log(x)*sin(x)
The derivative of the function/ log(x)*sin(x)

Derivative of log(x)*sin(x)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
log(x)*sin(x)
log⁡(x)sin⁡(x)\log{\left(x \right)} \sin{\left(x \right)} 
d                
--(log(x)*sin(x))
dx
ddxlog⁡(x)sin⁡(x)\frac{d}{d x} \log{\left(x \right)} \sin{\left(x \right)}
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result is: log⁡(x)cos⁡(x)+sin⁡(x)x\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}

The answer is:

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

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
sin(x)                
------ + cos(x)*log(x)
  x
log⁡(x)cos⁡(x)+sin⁡(x)x\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}
Simplify
The second derivative [src] 
  sin(x)                   2*cos(x)
- ------ - log(x)*sin(x) + --------
     2                        x    
    x
−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2- \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}
Simplify
The third derivative [src] 
                 3*sin(x)   3*cos(x)   2*sin(x)
-cos(x)*log(x) - -------- - -------- + --------
                    x           2          3   
                               x          x
−log⁡(x)cos⁡(x)−3sin⁡(x)x−3cos⁡(x)x2+2sin⁡(x)x3- \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}}
Simplify
The graph
Derivative of log(x)*sin(x)
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