Derivative of 3√ln(cos(4x+5))

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3*\/ log(cos(4*x + 5))

$$3 \sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}}$$

3*sqrt(log(cos(4*x + 5)))

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \log{\left(\cos{\left(4 x + 5 \right)} \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\cos{\left(4 x + 5 \right)} \right)}$ :
  1. Let $u = \cos{\left(4 x + 5 \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(4 x + 5 \right)}$ :
    1. Let $u = 4 x + 5$ .
    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 $\frac{d}{d x} \left(4 x + 5\right)$ :
      1. Differentiate $4 x + 5$ term by term:
        
        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: $4$
        
        The derivative of the constant $5$ is zero.
        
        The result is: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x + 5 \right)}$
    The result of the chain rule is:
    
    $- \frac{4 \sin{\left(4 x + 5 \right)}}{\cos{\left(4 x + 5 \right)}}$
  The result of the chain rule is:
  
  $- \frac{2 \sin{\left(4 x + 5 \right)}}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}} \cos{\left(4 x + 5 \right)}}$
So, the result is: $- \frac{6 \sin{\left(4 x + 5 \right)}}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}} \cos{\left(4 x + 5 \right)}}$
Now simplify:

$- \frac{6 \tan{\left(4 x + 5 \right)}}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}}}$

The answer is:

$- \frac{6 \tan{\left(4 x + 5 \right)}}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

         -6*sin(4*x + 5)          
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cos(4*x + 5)*\/ log(cos(4*x + 5))

$$- \frac{6 \sin{\left(4 x + 5 \right)}}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}} \cos{\left(4 x + 5 \right)}}$$

Simplify

The second derivative [src]

    /         2                        2                  \
    |    2*sin (5 + 4*x)            sin (5 + 4*x)         |
-12*|2 + --------------- + -------------------------------|
    |        2                2                           |
    \     cos (5 + 4*x)    cos (5 + 4*x)*log(cos(5 + 4*x))/
-----------------------------------------------------------
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                   \/ log(cos(5 + 4*x))

$$- \frac{12 \left(\frac{2 \sin^{2}{\left(4 x + 5 \right)}}{\cos^{2}{\left(4 x + 5 \right)}} + 2 + \frac{\sin^{2}{\left(4 x + 5 \right)}}{\log{\left(\cos{\left(4 x + 5 \right)} \right)} \cos^{2}{\left(4 x + 5 \right)}}\right)}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}}}$$

Simplify

The third derivative [src]

    /                             2                         2                                  2                 \             
    |            6           8*sin (5 + 4*x)           3*sin (5 + 4*x)                    6*sin (5 + 4*x)        |             
-24*|8 + ----------------- + --------------- + -------------------------------- + -------------------------------|*sin(5 + 4*x)
    |    log(cos(5 + 4*x))       2                2             2                    2                           |             
    \                         cos (5 + 4*x)    cos (5 + 4*x)*log (cos(5 + 4*x))   cos (5 + 4*x)*log(cos(5 + 4*x))/             
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                                               cos(5 + 4*x)*\/ log(cos(5 + 4*x))

$$- \frac{24 \left(\frac{8 \sin^{2}{\left(4 x + 5 \right)}}{\cos^{2}{\left(4 x + 5 \right)}} + 8 + \frac{6 \sin^{2}{\left(4 x + 5 \right)}}{\log{\left(\cos{\left(4 x + 5 \right)} \right)} \cos^{2}{\left(4 x + 5 \right)}} + \frac{6}{\log{\left(\cos{\left(4 x + 5 \right)} \right)}} + \frac{3 \sin^{2}{\left(4 x + 5 \right)}}{\log{\left(\cos{\left(4 x + 5 \right)} \right)}^{2} \cos^{2}{\left(4 x + 5 \right)}}\right) \sin{\left(4 x + 5 \right)}}{\sqrt{\log{\left(\cos{\left(4 x + 5 \right)} \right)}} \cos{\left(4 x + 5 \right)}}$$

Simplify

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Derivative of 3√ln(cos(4x+5))

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