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Identical expressions
y=x^ four *sinsinx
y equally x to the power of 4 multiply by sinus of sinus of x
y equally x to the power of four multiply by sinus of sinus of x
y=x4*sinsinx
y=x⁴*sinsinx
y=x^4sinsinx
y=x4sinsinx

The derivative of the function/ y=x^4*sinsinx

Derivative of y=x^4*sinsinx

The solution

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 4            
x *sin(sin(x))

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

x^4*sin(sin(x))

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

   3                4                   
4*x *sin(sin(x)) + x *cos(x)*cos(sin(x))

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

Simplify

The second derivative [src]

 2 /                  2 /   2                                    \                         \
x *\12*sin(sin(x)) - x *\cos (x)*sin(sin(x)) + cos(sin(x))*sin(x)/ + 8*x*cos(x)*cos(sin(x))/

$$x^{2} \left(- x^{2} \left(\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right) + 8 x \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 12 \sin{\left(\sin{\left(x \right)} \right)}\right)$$

Simplify

The third derivative [src]

  /                     2 /   2                                    \    3 /   2                                                    \                                 \
x*\24*sin(sin(x)) - 12*x *\cos (x)*sin(sin(x)) + cos(sin(x))*sin(x)/ - x *\cos (x)*cos(sin(x)) - 3*sin(x)*sin(sin(x)) + cos(sin(x))/*cos(x) + 36*x*cos(x)*cos(sin(x))/

$$x \left(- x^{3} \left(- 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} + \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - 12 x^{2} \left(\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right) + 36 x \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 24 \sin{\left(\sin{\left(x \right)} \right)}\right)$$

Simplify

The graph

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Identical expressions

Derivative of y=x^4*sinsinx

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Piecewise:

The solution