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

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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
           ____
   4      /  x 
sin (x)*\/  E
exsin⁡4(x)\sqrt{e^{x}} \sin^{4}{\left(x \right)}
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=sin⁡4(x)f{\left(x \right)} = \sin^{4}{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

      1. The derivative of sine is cosine:

        ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

      The result of the chain rule is:

      4sin⁡3(x)cos⁡(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}

    g(x)=exg{\left(x \right)} = \sqrt{e^{x}}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=exu = e^{x}.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddxex\frac{d}{d x} e^{x}:

      1. The derivative of exe^{x} is itself.

      The result of the chain rule is:

      ex22\frac{e^{\frac{x}{2}}}{2}

    The result is: ex2sin⁡4(x)2+4ex2sin⁡3(x)cos⁡(x)\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)}

  2. Now simplify:

    (sin⁡(x)+8cos⁡(x))ex2sin⁡3(x)2\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:

(sin⁡(x)+8cos⁡(x))ex2sin⁡3(x)2\frac{\left(\sin{\left(x \right)} + 8 \cos{\left(x \right)}\right) e^{\frac{x}{2}} \sin^{3}{\left(x \right)}}{2}

The graph
02468-8-6-4-2-1010-200200
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
ex2sin⁡4(x)2+4ex2sin⁡3(x)cos⁡(x)\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                       /
(−15sin⁡2(x)4+4sin⁡(x)cos⁡(x)+12cos⁡2(x))ex2sin⁡2(x)\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                                                                                            /
(−6(sin⁡2(x)−3cos⁡2(x))sin⁡(x)−8(5sin⁡2(x)−3cos⁡2(x))cos⁡(x)+sin⁡3(x)8+3sin⁡2(x)cos⁡(x))ex2sin⁡(x)\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
Derivative of y=sin^4sqrt(e^x)
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