Derivative of ln(sin(6x)+3x)+3x+3

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log(sin(6*x) + 3*x) + 3*x + 3

$$\left(3 x + \log{\left(3 x + \sin{\left(6 x \right)} \right)}\right) + 3$$

log(sin(6*x) + 3*x) + 3*x + 3

Detail solution

Differentiate $\left(3 x + \log{\left(3 x + \sin{\left(6 x \right)} \right)}\right) + 3$ term by term:
1. Differentiate $3 x + \log{\left(3 x + \sin{\left(6 x \right)} \right)}$ term by term:
  1. Let $u = 3 x + \sin{\left(6 x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + \sin{\left(6 x \right)}\right)$ :
    1. Differentiate $3 x + \sin{\left(6 x \right)}$ term by term:
      1. Let $u = 6 x$ .
      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 $\frac{d}{d x} 6 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $6$
        
        The result of the chain rule is:
        
        $6 \cos{\left(6 x \right)}$
      4. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      The result is: $6 \cos{\left(6 x \right)} + 3$
    The result of the chain rule is:
    
    $\frac{6 \cos{\left(6 x \right)} + 3}{3 x + \sin{\left(6 x \right)}}$
  4. 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 is: $3 + \frac{6 \cos{\left(6 x \right)} + 3}{3 x + \sin{\left(6 x \right)}}$
2. The derivative of the constant $3$ is zero.
The result is: $3 + \frac{6 \cos{\left(6 x \right)} + 3}{3 x + \sin{\left(6 x \right)}}$
Now simplify:

$\frac{3 \left(3 x + \sin{\left(6 x \right)} + 2 \cos{\left(6 x \right)} + 1\right)}{3 x + \sin{\left(6 x \right)}}$

The answer is:

$\frac{3 \left(3 x + \sin{\left(6 x \right)} + 2 \cos{\left(6 x \right)} + 1\right)}{3 x + \sin{\left(6 x \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

    3 + 6*cos(6*x)
3 + --------------
    sin(6*x) + 3*x

$$3 + \frac{6 \cos{\left(6 x \right)} + 3}{3 x + \sin{\left(6 x \right)}}$$

Simplify

The second derivative [src]

   /                             2\
   |             (1 + 2*cos(6*x)) |
-9*|4*sin(6*x) + -----------------|
   \               3*x + sin(6*x) /
-----------------------------------
           3*x + sin(6*x)

$$- \frac{9 \left(4 \sin{\left(6 x \right)} + \frac{\left(2 \cos{\left(6 x \right)} + 1\right)^{2}}{3 x + \sin{\left(6 x \right)}}\right)}{3 x + \sin{\left(6 x \right)}}$$

Simplify

The third derivative [src]

   /                              3                              \
   |              (1 + 2*cos(6*x))    6*(1 + 2*cos(6*x))*sin(6*x)|
54*|-4*cos(6*x) + ----------------- + ---------------------------|
   |                              2          3*x + sin(6*x)      |
   \              (3*x + sin(6*x))                               /
------------------------------------------------------------------
                          3*x + sin(6*x)

$$\frac{54 \left(- 4 \cos{\left(6 x \right)} + \frac{6 \left(2 \cos{\left(6 x \right)} + 1\right) \sin{\left(6 x \right)}}{3 x + \sin{\left(6 x \right)}} + \frac{\left(2 \cos{\left(6 x \right)} + 1\right)^{3}}{\left(3 x + \sin{\left(6 x \right)}\right)^{2}}\right)}{3 x + \sin{\left(6 x \right)}}$$

Simplify

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Derivative of ln(sin(6x)+3x)+3x+3

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The solution