Derivative of tan(2^x)ln(3x)

The solution

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   / x\         
tan\2 /*log(3*x)

$$\log{\left(3 x \right)} \tan{\left(2^{x} \right)}$$

tan(2^x)*log(3*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \tan{\left(2^{x} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(2^{x} \right)} = \frac{\sin{\left(2^{x} \right)}}{\cos{\left(2^{x} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(2^{x} \right)}$ and $g{\left(x \right)} = \cos{\left(2^{x} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2^{x}$ .
  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 $\frac{d}{d x} 2^{x}$ :
    1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
    The result of the chain rule is:
    
    $2^{x} \log{\left(2 \right)} \cos{\left(2^{x} \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2^{x}$ .
  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 $\frac{d}{d x} 2^{x}$ :
    1. $\frac{d}{d x} 2^{x} = 2^{x} \log{\left(2 \right)}$
    The result of the chain rule is:
    
    $- 2^{x} \log{\left(2 \right)} \sin{\left(2^{x} \right)}$
  Now plug in to the quotient rule:
  
  $\frac{2^{x} \log{\left(2 \right)} \sin^{2}{\left(2^{x} \right)} + 2^{x} \log{\left(2 \right)} \cos^{2}{\left(2^{x} \right)}}{\cos^{2}{\left(2^{x} \right)}}$
$g{\left(x \right)} = \log{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
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:
  
  $\frac{1}{x}$
The result is: $\frac{\left(2^{x} \log{\left(2 \right)} \sin^{2}{\left(2^{x} \right)} + 2^{x} \log{\left(2 \right)} \cos^{2}{\left(2^{x} \right)}\right) \log{\left(3 x \right)}}{\cos^{2}{\left(2^{x} \right)}} + \frac{\tan{\left(2^{x} \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   / x\                                    
tan\2 /    x /       2/ x\\                
------- + 2 *\1 + tan \2 //*log(2)*log(3*x)
   x

$$2^{x} \left(\tan^{2}{\left(2^{x} \right)} + 1\right) \log{\left(2 \right)} \log{\left(3 x \right)} + \frac{\tan{\left(2^{x} \right)}}{x}$$

Simplify

The second derivative [src]

     / x\      x /       2/ x\\                                                               
  tan\2 /   2*2 *\1 + tan \2 //*log(2)    x    2    /       2/ x\\ /       x    / x\\         
- ------- + -------------------------- + 2 *log (2)*\1 + tan \2 //*\1 + 2*2 *tan\2 //*log(3*x)
      2                 x                                                                     
     x

$$2^{x} \left(2 \cdot 2^{x} \tan{\left(2^{x} \right)} + 1\right) \left(\tan^{2}{\left(2^{x} \right)} + 1\right) \log{\left(2 \right)}^{2} \log{\left(3 x \right)} + \frac{2 \cdot 2^{x} \left(\tan^{2}{\left(2^{x} \right)} + 1\right) \log{\left(2 \right)}}{x} - \frac{\tan{\left(2^{x} \right)}}{x^{2}}$$

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of tan(2^x)ln(3x)

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