Derivative of -(sin(lnx)+cos(lnx))/x

The solution

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

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

(-sin(log(x)) - cos(log(x)))/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = - \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \log{\left(x \right)}$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
    So, the result is: $\frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \log{\left(x \right)}$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result of the chain rule is:
      
      $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
    So, the result is: $- \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
  The result is: $\frac{\sin{\left(\log{\left(x \right)} \right)}}{x} - \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

2*(-5*cos(log(x)) + 5*sin(log(x)))
----------------------------------
                 4                
                x

$$\frac{2 \left(5 \sin{\left(\log{\left(x \right)} \right)} - 5 \cos{\left(\log{\left(x \right)} \right)}\right)}{x^{4}}$$

Simplify

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Derivative of -(sin(lnx)+cos(lnx))/x

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The solution