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Identical expressions
y=e^x^2sin^4x
y equally e to the power of x squared sinus of to the power of 4x
y=ex2sin4x
y=e^x²sin⁴x
y=e to the power of x to the power of 2sin to the power of 4x

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

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

The solution

You have entered [src]

 / 2\        
 \x /    4   
E    *sin (x)

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

E^(x^2)*sin(x)^4

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

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The first derivative [src]

             / 2\                     / 2\
       4     \x /        3            \x /
2*x*sin (x)*e     + 4*sin (x)*cos(x)*e

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

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

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

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)

$$4 \left(x \left(2 x^{2} + 3\right) \sin^{3}{\left(x \right)} - 6 x \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} + 6 \left(2 x^{2} + 1\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 2 \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}\right) e^{x^{2}} \sin{\left(x \right)}$$

Simplify

The graph

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Identical expressions

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

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The solution