Derivative of cos(2x)*ln(x)

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cos(2*x)*log(x)

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

cos(2*x)*log(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \cos{\left(2 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
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 $\frac{d}{d x} 2 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $2$
  The result of the chain rule is:
  
  $- 2 \sin{\left(2 x \right)}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $- 2 \log{\left(x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{x}$

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

 /cos(2*x)   4*sin(2*x)                    \
-|-------- + ---------- + 4*cos(2*x)*log(x)|
 |    2          x                         |
 \   x                                     /

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

Simplify

The third derivative [src]

  /cos(2*x)   6*cos(2*x)   3*sin(2*x)                    \
2*|-------- - ---------- + ---------- + 4*log(x)*sin(2*x)|
  |    3          x             2                        |
  \   x                        x                         /

$$2 \left(4 \log{\left(x \right)} \sin{\left(2 x \right)} - \frac{6 \cos{\left(2 x \right)}}{x} + \frac{3 \sin{\left(2 x \right)}}{x^{2}} + \frac{\cos{\left(2 x \right)}}{x^{3}}\right)$$

Simplify

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Derivative of cos(2x)*ln(x)

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Piecewise:

The solution