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Identical expressions
(e^sin(x)*tg(x))
(e to the power of sinus of (x) multiply by tg(x))
(esin(x)*tg(x))
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(e^sin(x)tg(x))
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Similar expressions
(e^sinx*tg(x))

The derivative of the function/ (e^sin(x)*tg(x))

Derivative of (e^sin(x)*tg(x))

The solution

You have entered [src]

 sin(x)       
E      *tan(x)

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

E^sin(x)*tan(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = e^{\sin{\left(x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $e^{\sin{\left(x \right)}} \cos{\left(x \right)}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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 is: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\sin{\left(x \right)}}}{\cos^{2}{\left(x \right)}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} \tan{\left(x \right)}$
Now simplify:

$\frac{\left(- \sin^{3}{\left(x \right)} + \sin{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}}}{\cos^{2}{\left(x \right)}}$

The answer is:

$\frac{\left(- \sin^{3}{\left(x \right)} + \sin{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}}}{\cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (e^sin(x)*tg(x))

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Piecewise:

The solution