Derivative of 7sin^2((pi/16)*t)-3t

1. Differentiate $- 3 t + 7 \sin^{2}{\left(t \frac{\pi}{16} \right)}$ term by term:

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

      1. Let $u = \sin{\left(t \frac{\pi}{16} \right)}$.

      2. Apply the power rule: $u^{2}$ goes to $2 u$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d t} \sin{\left(t \frac{\pi}{16} \right)}$:

         1. Let $u = t \frac{\pi}{16}$.

         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 $\frac{d}{d t} t \frac{\pi}{16}$:

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

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

               So, the result is: $\frac{\pi}{16}$

            The result of the chain rule is:

            $$\frac{\pi \cos{\left(t \frac{\pi}{16} \right)}}{16}$$

         The result of the chain rule is:

         $$\frac{\pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8}$$

      So, the result is: $\frac{7 \pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8}$

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

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

      So, the result is: $-3$

   The result is: $\frac{7 \pi \sin{\left(t \frac{\pi}{16} \right)} \cos{\left(t \frac{\pi}{16} \right)}}{8} - 3$

2. Now simplify:

   $$\frac{7 \pi \sin{\left(\frac{\pi t}{8} \right)}}{16} - 3$$

The answer is: