Derivative of cos(3x)*log(x)

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cos(3*x)*log(x)

$$\log{\left(x \right)} \cos{\left(3 x \right)}$$

d                  
--(cos(3*x)*log(x))
dx

$$\frac{d}{d x} \log{\left(x \right)} \cos{\left(3 x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \cos{\left(3 x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x$ .
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 $\frac{d}{d x} 3 x$ :
  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: $3$
  The result of the chain rule is:
  
  $- 3 \sin{\left(3 x \right)}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x}$

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

 /cos(3*x)   6*sin(3*x)                    \
-|-------- + ---------- + 9*cos(3*x)*log(x)|
 |    2          x                         |
 \   x                                     /

$$- (9 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{6 \sin{\left(3 x \right)}}{x} + \frac{\cos{\left(3 x \right)}}{x^{2}})$$

Simplify

The third derivative [src]

  27*cos(3*x)   2*cos(3*x)   9*sin(3*x)                     
- ----------- + ---------- + ---------- + 27*log(x)*sin(3*x)
       x             3            2                         
                    x            x

$$27 \log{\left(x \right)} \sin{\left(3 x \right)} - \frac{27 \cos{\left(3 x \right)}}{x} + \frac{9 \sin{\left(3 x \right)}}{x^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{3}}$$

Simplify

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Derivative of cos(3x)*log(x)

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The solution