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Derivative of x*sin(4*x)
Identical expressions
x^ two * eight *cos(four *x)/ five
x squared multiply by 8 multiply by co sinus of e of (4 multiply by x) divide by 5
x to the power of two multiply by eight multiply by co sinus of e of (four multiply by x) divide by five
x2*8*cos(4*x)/5
x2*8*cos4*x/5
x²*8*cos(4*x)/5
x to the power of 2*8*cos(4*x)/5
x^28cos(4x)/5
x28cos(4x)/5
x28cos4x/5
x^28cos4x/5
x^2*8*cos(4*x) divide by 5

The derivative of the function/ x^2*8*cos(4*x)/5

Derivative of x^28cos(4*x)/5

The solution

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 2           
x *8*cos(4*x)
-------------
      5

$$\frac{8 x^{2} \cos{\left(4 x \right)}}{5}$$

((x^2*8)*cos(4*x))/5

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. 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 $2 x$
    $g{\left(x \right)} = \cos{\left(4 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 4 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    The result is: $- 4 x^{2} \sin{\left(4 x \right)} + 2 x \cos{\left(4 x \right)}$
  So, the result is: $- 32 x^{2} \sin{\left(4 x \right)} + 16 x \cos{\left(4 x \right)}$
So, the result is: $- \frac{32 x^{2} \sin{\left(4 x \right)}}{5} + \frac{16 x \cos{\left(4 x \right)}}{5}$
Now simplify:

$\frac{16 x \left(- 2 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right)}{5}$

The answer is:

$\frac{16 x \left(- 2 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right)}{5}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2                         
  32*x *sin(4*x)   16*x*cos(4*x)
- -------------- + -------------
        5                5

$$- \frac{32 x^{2} \sin{\left(4 x \right)}}{5} + \frac{16 x \cos{\left(4 x \right)}}{5}$$

Simplify

The second derivative [src]

   /                   2                    \
16*\-8*x*sin(4*x) - 8*x *cos(4*x) + cos(4*x)/
---------------------------------------------
                      5

$$\frac{16 \left(- 8 x^{2} \cos{\left(4 x \right)} - 8 x \sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right)}{5}$$

Simplify

The third derivative [src]

   /                                 2         \
64*\-3*sin(4*x) - 12*x*cos(4*x) + 8*x *sin(4*x)/
------------------------------------------------
                       5

$$\frac{64 \left(8 x^{2} \sin{\left(4 x \right)} - 12 x \cos{\left(4 x \right)} - 3 \sin{\left(4 x \right)}\right)}{5}$$

Simplify

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Derivative of x^28cos(4*x)/5

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Piecewise:

The solution

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Identical expressions

Derivative of x^2*8*cos(4*x)/5

The graph:

Piecewise:

The solution

Derivative of x^28cos(4*x)/5