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y=e^x^2sin^4x
The derivative of the function/ y=e^x^2sin^4x

Derivative of y=e^x^2sin^4x

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

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The solution

You have entered [src] 
 / 2\        
 \x /    4   
E    *sin (x)
ex2sin⁡4(x)e^{x^{2}} \sin^{4}{\left(x \right)} 
E^(x^2)*sin(x)^4
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)=ex2f{\left(x \right)} = e^{x^{2}}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=x2u = x^{2}.

    2. The derivative of eue^{u} is itself.

    3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} x^{2}:

      1. Apply the power rule: x2x^{2} goes to 2x2 x

      The result of the chain rule is:

      2xex22 x e^{x^{2}}

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

    The result is: 2xex2sin⁡4(x)+4ex2sin⁡3(x)cos⁡(x)2 x e^{x^{2}} \sin^{4}{\left(x \right)} + 4 e^{x^{2}} \sin^{3}{\left(x \right)} \cos{\left(x \right)}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-10101e44-5e43
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
             / 2\                     / 2\
       4     \x /        3            \x /
2*x*sin (x)*e     + 4*sin (x)*cos(x)*e
2xex2sin⁡4(x)+4ex2sin⁡3(x)cos⁡(x)2 x e^{x^{2}} \sin^{4}{\left(x \right)} + 4 e^{x^{2}} \sin^{3}{\left(x \right)} \cos{\left(x \right)}
Simplify
The second derivative [src] 
                                                                              / 2\
     2    /       2           2         2    /       2\                    \  \x /
2*sin (x)*\- 2*sin (x) + 6*cos (x) + sin (x)*\1 + 2*x / + 8*x*cos(x)*sin(x)/*e
2(8xsin⁡(x)cos⁡(x)+(2x2+1)sin⁡2(x)−2sin⁡2(x)+6cos⁡2(x))ex2sin⁡2(x)2 \left(8 x \sin{\left(x \right)} \cos{\left(x \right)} + \left(2 x^{2} + 1\right) \sin^{2}{\left(x \right)} - 2 \sin^{2}{\left(x \right)} + 6 \cos^{2}{\left(x \right)}\right) e^{x^{2}} \sin^{2}{\left(x \right)}
Simplify
The third derivative [src] 
                                                                                                                                  / 2\       
  /    /       2           2   \               3    /       2\       /   2           2   \               2    /       2\       \  \x /       
4*\- 2*\- 3*cos (x) + 5*sin (x)/*cos(x) + x*sin (x)*\3 + 2*x / - 6*x*\sin (x) - 3*cos (x)/*sin(x) + 6*sin (x)*\1 + 2*x /*cos(x)/*e    *sin(x)
4(x(2x2+3)sin⁡3(x)−6x(sin⁡2(x)−3cos⁡2(x))sin⁡(x)+6(2x2+1)sin⁡2(x)cos⁡(x)−2(5sin⁡2(x)−3cos⁡2(x))cos⁡(x))ex2sin⁡(x)4 \left(x \left(2 x^{2} + 3\right) \sin^{3}{\left(x \right)} - 6 x \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} + 6 \left(2 x^{2} + 1\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 2 \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}\right) e^{x^{2}} \sin{\left(x \right)}
Simplify
The graph
Derivative of y=e^x^2sin^4x
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