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Identical expressions
y=sin^4sqrt(e^x)
y equally sinus of to the power of 4 square root of (e to the power of x)
y=sin^4√(e^x)
y=sin4sqrt(ex)
y=sin4sqrtex
y=sin⁴sqrt(e^x)
y=sin^4sqrte^x

The derivative of the function/ y=sin^4sqrt(e^x)

Derivative of y=sin^4sqrt(e^x)

The solution

You have entered [src]

           ____
   4      /  x 
sin (x)*\/  E

$$\sqrt{e^{x}} \sin^{4}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sin^{4}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
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:
  
  $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
$g{\left(x \right)} = \sqrt{e^{x}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = e^{x}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{x}$ :
  1. The derivative of $e^{x}$ is itself.
  The result of the chain rule is:
  
  $\frac{e^{\frac{x}{2}}}{2}$
The result is: $\frac{e^{\frac{x}{2}} \sin^{4}{\left(x \right)}}{2} + 4 e^{\frac{x}{2}} \sin^{3}{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         x                      
         -                     x
   4     2                     -
sin (x)*e         3            2
---------- + 4*sin (x)*cos(x)*e 
    2

$$\frac{e^{\frac{x}{2}} \sin^{4}{\left(x \right)}}{2} + 4 e^{\frac{x}{2}} \sin^{3}{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The second derivative [src]

                                                     x
        /                   2                     \  -
   2    |      2      15*sin (x)                  |  2
sin (x)*|12*cos (x) - ---------- + 4*cos(x)*sin(x)|*e 
        \                 4                       /

$$\left(- \frac{15 \sin^{2}{\left(x \right)}}{4} + 4 \sin{\left(x \right)} \cos{\left(x \right)} + 12 \cos^{2}{\left(x \right)}\right) e^{\frac{x}{2}} \sin^{2}{\left(x \right)}$$

Simplify

The third derivative [src]

                                                                                                    x       
/   3                                                                                            \  -       
|sin (x)     /       2           2   \            /   2           2   \               2          |  2       
|------- - 8*\- 3*cos (x) + 5*sin (x)/*cos(x) - 6*\sin (x) - 3*cos (x)/*sin(x) + 3*sin (x)*cos(x)|*e *sin(x)
\   8                                                                                            /

$$\left(- 6 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} - 8 \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} + \frac{\sin^{3}{\left(x \right)}}{8} + 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right) e^{\frac{x}{2}} \sin{\left(x \right)}$$

Simplify

The graph

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Derivative of y=sin^4sqrt(e^x)

The graph:

Piecewise:

The solution