Derivative of e^tan(x)

The solution

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 tan(x)
e

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

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

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

Transform

Detail solution

Let $u = \tan{\left(x \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of e^tan(x)

The graph:

Piecewise:

The solution