Derivative of y=x(sin(ln(x))-cos(ln(x)))

The solution

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}$ term by term:
  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:
    
    $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
  4. 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: $\frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
  The result is: $\frac{\sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
The result is: $x \left(\frac{\sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\right) + \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of y=x(sin(ln(x))-cos(ln(x)))

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Derivative of y=x(sin(ln(x))-cos(ln(x)))

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The solution