Derivative of 2^tg(3x)

The solution

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 tan(3*x)
2

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

2^tan(3*x)

Detail solution

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

$\frac{2^{\tan{\left(3 x \right)}} \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(3 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   tan(3*x) /       2     \ /             /       2     \       \       
9*2        *\1 + tan (3*x)/*\2*tan(3*x) + \1 + tan (3*x)/*log(2)/*log(2)

$$9 \cdot 2^{\tan{\left(3 x \right)}} \left(\left(\tan^{2}{\left(3 x \right)} + 1\right) \log{\left(2 \right)} + 2 \tan{\left(3 x \right)}\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \log{\left(2 \right)}$$

Simplify

The third derivative [src]

                             /                                 2                                            \       
    tan(3*x) /       2     \ |         2        /       2     \     2        /       2     \                |       
27*2        *\1 + tan (3*x)/*\2 + 6*tan (3*x) + \1 + tan (3*x)/ *log (2) + 6*\1 + tan (3*x)/*log(2)*tan(3*x)/*log(2)

$$27 \cdot 2^{\tan{\left(3 x \right)}} \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(\left(\tan^{2}{\left(3 x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} + 6 \left(\tan^{2}{\left(3 x \right)} + 1\right) \log{\left(2 \right)} \tan{\left(3 x \right)} + 6 \tan^{2}{\left(3 x \right)} + 2\right) \log{\left(2 \right)}$$

Simplify

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Identical expressions

Derivative of 2^tg(3x)

The graph:

Piecewise:

The solution