Derivative of y=tgx/2^x

The solution

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tan(x)
------
   x  
  2

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

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Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \tan{\left(x \right)}$ and $g{\left(x \right)} = 2^{x}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=tgx/2^x

The graph:

Piecewise:

The solution