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Derivative of:
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Integral of d{x}:
x^2sinx^2
Identical expressions
y=x^ two sinx^2
y equally x squared sinus of x squared
y equally x to the power of two sinus of x squared
y=x2sinx2
y=x²sinx²
y=x to the power of 2sinx to the power of 2
Similar expressions
(-1)/(cos(x)^2*sin(x)^(2*x))
(pi^2-x^2)(sin(x))^2

The derivative of the function/ y=x^2sinx^2

Derivative of y=x^2sinx^2

The solution

You have entered [src]

 2    2   
x *sin (x)

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

d / 2    2   \
--\x *sin (x)/
dx

$$\frac{d}{d x} x^{2} \sin^{2}{\left(x \right)}$$

Transform

Detail solution

Apply the product rule:

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

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

$x \left(x \sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 1\right)$

The answer is:

$x \left(x \sin{\left(2 x \right)} - \cos{\left(2 x \right)} + 1\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2         2              
2*x*sin (x) + 2*x *cos(x)*sin(x)

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

Simplify

The second derivative [src]

  /   2       2 /   2         2   \                    \
2*\sin (x) - x *\sin (x) - cos (x)/ + 4*x*cos(x)*sin(x)/

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=x^2sinx^2

The graph:

Piecewise:

The solution