Derivative of tan(x)/e^x

The solution

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tan(x)
------
   x  
  e

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

d /tan(x)\
--|------|
dx|   x  |
  \  e   /

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

Transform

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 $g{\left(x \right)} = e^{x}$ .

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. The derivative of $e^{x}$ is itself.
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of tan(x)/e^x

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The solution