Derivative of 3x^2sin4x+4x^3cos4x

1. Differentiate $4 x^{3} \cos{\left(4 x \right)} + 3 x^{2} \sin{\left(4 x \right)}$ term by term:

   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 $2 x$

         $g{\left(x \right)} = \sin{\left(4 x \right)}$; 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: $4 x^{2} \cos{\left(4 x \right)} + 2 x \sin{\left(4 x \right)}$

      So, the result is: $12 x^{2} \cos{\left(4 x \right)} + 6 x \sin{\left(4 x \right)}$

   2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the product rule:

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

         $f{\left(x \right)} = x^{3}$; to find $\frac{d}{d x} f{\left(x \right)}$:

         1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

         $g{\left(x \right)} = \cos{\left(4 x \right)}$; 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.

               1. 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: $- 4 x^{3} \sin{\left(4 x \right)} + 3 x^{2} \cos{\left(4 x \right)}$

      So, the result is: $- 16 x^{3} \sin{\left(4 x \right)} + 12 x^{2} \cos{\left(4 x \right)}$

   The result is: $- 16 x^{3} \sin{\left(4 x \right)} + 24 x^{2} \cos{\left(4 x \right)} + 6 x \sin{\left(4 x \right)}$

2. Now simplify:

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

The answer is:

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