Derivative of lnx*tg4x

The solution

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log(x)*tan(4*x)

$$\log{\left(x \right)} \tan{\left(4 x \right)}$$

log(x)*tan(4*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \tan{\left(4 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(4 x \right)}$ and $g{\left(x \right)} = \cos{\left(4 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 4 x$ .
  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 $\frac{d}{d x} 4 x$ :
    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: $4$
    The result of the chain rule is:
    
    $4 \cos{\left(4 x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 4 x$ .
  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 $\frac{d}{d x} 4 x$ :
    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: $4$
    The result of the chain rule is:
    
    $- 4 \sin{\left(4 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}$
The result is: $\frac{\left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(4 x \right)}} + \frac{\tan{\left(4 x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

tan(4*x)   /         2     \       
-------- + \4 + 4*tan (4*x)/*log(x)
   x

$$\left(4 \tan^{2}{\left(4 x \right)} + 4\right) \log{\left(x \right)} + \frac{\tan{\left(4 x \right)}}{x}$$

Simplify

The second derivative [src]

               /       2     \                                     
  tan(4*x)   8*\1 + tan (4*x)/      /       2     \                
- -------- + ----------------- + 32*\1 + tan (4*x)/*log(x)*tan(4*x)
      2              x                                             
     x

$$32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(x \right)} \tan{\left(4 x \right)} + \frac{8 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{x} - \frac{\tan{\left(4 x \right)}}{x^{2}}$$

Simplify

The third derivative [src]

  /             /       2     \      /       2     \                                                       \
  |tan(4*x)   6*\1 + tan (4*x)/   48*\1 + tan (4*x)/*tan(4*x)      /       2     \ /         2     \       |
2*|-------- - ----------------- + --------------------------- + 64*\1 + tan (4*x)/*\1 + 3*tan (4*x)/*log(x)|
  |    3               2                       x                                                           |
  \   x               x                                                                                    /

$$2 \left(64 \left(\tan^{2}{\left(4 x \right)} + 1\right) \left(3 \tan^{2}{\left(4 x \right)} + 1\right) \log{\left(x \right)} + \frac{48 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)}}{x} - \frac{6 \left(\tan^{2}{\left(4 x \right)} + 1\right)}{x^{2}} + \frac{\tan{\left(4 x \right)}}{x^{3}}\right)$$

Simplify

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Derivative of lnx*tg4x

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The solution