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y=x^ two ×(2cosx-sinx)
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y=x^2×(2cosx+sinx)

The derivative of the function/ y=x^2×(2cosx-sinx)

Derivative of y=x^2×(2cosx-sinx)

The solution

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 2                    
x *(2*cos(x) - sin(x))

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

x^2*(2*cos(x) - sin(x))

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
$g{\left(x \right)} = - \sin{\left(x \right)} + 2 \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $- \sin{\left(x \right)} + 2 \cos{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 2 \sin{\left(x \right)}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $- \cos{\left(x \right)}$
  The result is: $- 2 \sin{\left(x \right)} - \cos{\left(x \right)}$
The result is: $x^{2} \left(- 2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) + 2 x \left(- \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                         2                                               
-12*sin(x) - 6*cos(x) + x *(2*sin(x) + cos(x)) + 6*x*(-2*cos(x) + sin(x))

$$x^{2} \left(2 \sin{\left(x \right)} + \cos{\left(x \right)}\right) + 6 x \left(\sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) - 12 \sin{\left(x \right)} - 6 \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of y=x^2×(2cosx-sinx)

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The solution