Derivative of y=x^2sin4x

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 2         
x *sin(4*x)

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

x^2*sin(4*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 $2 x$
$g{\left(x \right)} = \sin{\left(4 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x$ .
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 $\frac{d}{d x} 4 x$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $4$
  The result of the chain rule is:
  
  $4 \cos{\left(4 x \right)}$
The result is: $4 x^{2} \cos{\left(4 x \right)} + 2 x \sin{\left(4 x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph