Derivative of 2^(tg(4x+5))

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 tan(4*x + 5)
2

2^{\tan{\left(4 x + 5 \right)}}

2^tan(4*x + 5)

Detail solution

Let $u = \tan{\left(4 x + 5 \right)}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(4 x + 5 \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(4 x + 5 \right)} = \frac{\sin{\left(4 x + 5 \right)}}{\cos{\left(4 x + 5 \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(4 x + 5 \right)}$ and $g{\left(x \right)} = \cos{\left(4 x + 5 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 4 x + 5$ .
  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 $\frac{d}{d x} \left(4 x + 5\right)$ :
    1. Differentiate $4 x + 5$ term by term:
      1. 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: $4$
      2. The derivative of the constant $5$ is zero.
      The result is: $4$
    The result of the chain rule is:
    
    $4 \cos{\left(4 x + 5 \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
      1. 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: $4$
      2. The derivative of the constant $5$ is zero.
      The result is: $4$
    The result of the chain rule is:
    
    $- 4 \sin{\left(4 x + 5 \right)}$
  Now plug in to the quotient rule:
  
  $\frac{4 \sin^{2}{\left(4 x + 5 \right)} + 4 \cos^{2}{\left(4 x + 5 \right)}}{\cos^{2}{\left(4 x + 5 \right)}}$
The result of the chain rule is:

$\frac{2^{\tan{\left(4 x + 5 \right)}} \left(4 \sin^{2}{\left(4 x + 5 \right)} + 4 \cos^{2}{\left(4 x + 5 \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(4 x + 5 \right)}}$
Now simplify:

$\frac{2^{\tan{\left(4 x + 5 \right)} + 2} \log{\left(2 \right)}}{\cos^{2}{\left(4 x + 5 \right)}}$

The answer is:

\frac{2^{\tan{\left(4 x + 5 \right)} + 2} \log{\left(2 \right)}}{\cos^{2}{\left(4 x + 5 \right)}}

The graph

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Plot the graph f'(x)

The first derivative [src]

 tan(4*x + 5) /         2         \       
2            *\4 + 4*tan (4*x + 5)/*log(2)

2^{\tan{\left(4 x + 5 \right)}} \left(4 \tan^{2}{\left(4 x + 5 \right)} + 4\right) \log{\left(2 \right)}

Simplify

The second derivative [src]

    tan(5 + 4*x) /       2         \ /                 /       2         \       \       
16*2            *\1 + tan (5 + 4*x)/*\2*tan(5 + 4*x) + \1 + tan (5 + 4*x)/*log(2)/*log(2)

16 \cdot 2^{\tan{\left(4 x + 5 \right)}} \left(\left(\tan^{2}{\left(4 x + 5 \right)} + 1\right) \log{\left(2 \right)} + 2 \tan{\left(4 x + 5 \right)}\right) \left(\tan^{2}{\left(4 x + 5 \right)} + 1\right) \log{\left(2 \right)}

Simplify

The third derivative [src]

                                     /                                         2                                                    \       
    tan(5 + 4*x) /       2         \ |         2            /       2         \     2        /       2         \                    |       
64*2            *\1 + tan (5 + 4*x)/*\2 + 6*tan (5 + 4*x) + \1 + tan (5 + 4*x)/ *log (2) + 6*\1 + tan (5 + 4*x)/*log(2)*tan(5 + 4*x)/*log(2)

64 \cdot 2^{\tan{\left(4 x + 5 \right)}} \left(\tan^{2}{\left(4 x + 5 \right)} + 1\right) \left(\left(\tan^{2}{\left(4 x + 5 \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} + 6 \left(\tan^{2}{\left(4 x + 5 \right)} + 1\right) \log{\left(2 \right)} \tan{\left(4 x + 5 \right)} + 6 \tan^{2}{\left(4 x + 5 \right)} + 2\right) \log{\left(2 \right)}

Simplify

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Derivative of 2^(tg(4x+5))

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