Derivative of (-pi*sin((pi*x)/4))/4-20

1. Differentiate $\frac{- \pi \sin{\left(\frac{\pi x}{4} \right)}}{4} - 20$ term by term:

   1. 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. Let $u = \frac{\pi x}{4}$.

         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 $\frac{d}{d x} \frac{\pi x}{4}$:

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

                  So, the result is: $\pi$

               So, the result is: $\frac{\pi}{4}$

            The result of the chain rule is:

            $$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{4}$$

         So, the result is: $- \frac{\pi^{2} \cos{\left(\frac{\pi x}{4} \right)}}{4}$

      So, the result is: $- \frac{\pi^{2} \cos{\left(\frac{\pi x}{4} \right)}}{16}$

   2. The derivative of the constant $-20$ is zero.

   The result is: $- \frac{\pi^{2} \cos{\left(\frac{\pi x}{4} \right)}}{16}$

2. Now simplify:

   $$- \frac{\pi^{2} \cos{\left(\frac{\pi x}{4} \right)}}{16}$$

The answer is:

$$- \frac{\pi^{2} \cos{\left(\frac{\pi x}{4} \right)}}{16}$$