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Identical expressions
e^(sinx)*cos^2x
e to the power of ( sinus of x) multiply by co sinus of e of squared x
e(sinx)*cos2x
esinx*cos2x
e^(sinx)*cos²x
e to the power of (sinx)*cos to the power of 2x
e^(sinx)cos^2x
e(sinx)cos2x
esinxcos2x
e^sinxcos^2x

The derivative of the function/ e^(sinx)*cos^2x

Derivative of e^(sinx)*cos^2x

The solution

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 sin(x)    2   
E      *cos (x)

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

E^sin(x)*cos(x)^2

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3     sin(x)             sin(x)       
cos (x)*e       - 2*cos(x)*e      *sin(x)

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

Simplify

The second derivative [src]

/       2           2         2    /     2            \        2          \  sin(x)
\- 2*cos (x) + 2*sin (x) - cos (x)*\- cos (x) + sin(x)/ - 4*cos (x)*sin(x)/*e

$$\left(- \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + 2 \sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) e^{\sin{\left(x \right)}}$$

Simplify

The third derivative [src]

/       2           2                    2    /       2              \     /     2            \       \         sin(x)
\- 6*cos (x) + 6*sin (x) + 8*sin(x) - cos (x)*\1 - cos (x) + 3*sin(x)/ + 6*\- cos (x) + sin(x)/*sin(x)/*cos(x)*e

$$\left(6 \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} - \left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} + 6 \sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} - 6 \cos^{2}{\left(x \right)}\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)}$$

Simplify

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Identical expressions

Derivative of e^(sinx)*cos^2x

The graph:

Piecewise:

The solution