Derivative of ln(2x+cos(5x))

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log(2*x + cos(5*x))

$$\log{\left(2 x + \cos{\left(5 x \right)} \right)}$$

log(2*x + cos(5*x))

Detail solution

Let $u = 2 x + \cos{\left(5 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + \cos{\left(5 x \right)}\right)$ :
1. Differentiate $2 x + \cos{\left(5 x \right)}$ term by term:
  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: $2$
  2. Let $u = 5 x$ .
  3. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x \right)}$
  The result is: $2 - 5 \sin{\left(5 x \right)}$
The result of the chain rule is:

$\frac{2 - 5 \sin{\left(5 x \right)}}{2 x + \cos{\left(5 x \right)}}$

The answer is:

$\frac{2 - 5 \sin{\left(5 x \right)}}{2 x + \cos{\left(5 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{2 - 5 \sin{\left(5 x \right)}}{2 x + \cos{\left(5 x \right)}}$$

Simplify

The second derivative [src]

 /                               2\ 
 |              (-2 + 5*sin(5*x)) | 
-|25*cos(5*x) + ------------------| 
 \                2*x + cos(5*x)  / 
------------------------------------
           2*x + cos(5*x)

$$- \frac{25 \cos{\left(5 x \right)} + \frac{\left(5 \sin{\left(5 x \right)} - 2\right)^{2}}{2 x + \cos{\left(5 x \right)}}}{2 x + \cos{\left(5 x \right)}}$$

Simplify

The third derivative [src]

                                  3                                
               2*(-2 + 5*sin(5*x))    75*(-2 + 5*sin(5*x))*cos(5*x)
125*sin(5*x) - -------------------- - -----------------------------
                                2             2*x + cos(5*x)       
                (2*x + cos(5*x))                                   
-------------------------------------------------------------------
                           2*x + cos(5*x)

$$\frac{125 \sin{\left(5 x \right)} - \frac{75 \left(5 \sin{\left(5 x \right)} - 2\right) \cos{\left(5 x \right)}}{2 x + \cos{\left(5 x \right)}} - \frac{2 \left(5 \sin{\left(5 x \right)} - 2\right)^{3}}{\left(2 x + \cos{\left(5 x \right)}\right)^{2}}}{2 x + \cos{\left(5 x \right)}}$$

Simplify

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Derivative of ln(2x+cos(5x))

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The solution