Derivative of log(x)*cos(x)

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log(x)*cos(x)

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

log(x)*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of log(x)*cos(x)

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The solution