Derivative of y'''=-8cos(x/2)+xsin(x/2)-48x+12

1. Differentiate $\left(- 48 x + \left(x \sin{\left(\frac{x}{2} \right)} - 8 \cos{\left(\frac{x}{2} \right)}\right)\right) + 12$ term by term:

   1. Differentiate $- 48 x + \left(x \sin{\left(\frac{x}{2} \right)} - 8 \cos{\left(\frac{x}{2} \right)}\right)$ term by term:

      1. Differentiate $x \sin{\left(\frac{x}{2} \right)} - 8 \cos{\left(\frac{x}{2} \right)}$ term by term:

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

            1. Let $u = \frac{x}{2}$.

            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 $\frac{d}{d x} \frac{x}{2}$:

               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: $\frac{1}{2}$

               The result of the chain rule is:

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

            So, the result is: $4 \sin{\left(\frac{x}{2} \right)}$

         2. Apply the product rule:

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

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

            1. Apply the power rule: $x$ goes to $1$

            $g{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

            1. Let $u = \frac{x}{2}$.

            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 $\frac{d}{d x} \frac{x}{2}$:

               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: $\frac{1}{2}$

               The result of the chain rule is:

               $$\frac{\cos{\left(\frac{x}{2} \right)}}{2}$$

            The result is: $\frac{x \cos{\left(\frac{x}{2} \right)}}{2} + \sin{\left(\frac{x}{2} \right)}$

         The result is: $\frac{x \cos{\left(\frac{x}{2} \right)}}{2} + 5 \sin{\left(\frac{x}{2} \right)}$

      2. 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: $-48$

      The result is: $\frac{x \cos{\left(\frac{x}{2} \right)}}{2} + 5 \sin{\left(\frac{x}{2} \right)} - 48$

   2. The derivative of the constant $12$ is zero.

   The result is: $\frac{x \cos{\left(\frac{x}{2} \right)}}{2} + 5 \sin{\left(\frac{x}{2} \right)} - 48$

2. Now simplify:

   $$\frac{x \cos{\left(\frac{x}{2} \right)}}{2} + 5 \sin{\left(\frac{x}{2} \right)} - 48$$

The answer is:

$$\frac{x \cos{\left(\frac{x}{2} \right)}}{2} + 5 \sin{\left(\frac{x}{2} \right)} - 48$$