Derivative of cos(pi*x)-sin(pi*x)

1. Differentiate $- \sin{\left(\pi x \right)} + \cos{\left(\pi x \right)}$ term by term:

   1. Let $u = \pi x$.

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

      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: $\pi$

      The result of the chain rule is:

      $$- \pi \sin{\left(\pi x \right)}$$

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

      1. Let $u = \pi x$.

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

         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: $\pi$

         The result of the chain rule is:

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

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

   The result is: $- \pi \sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}$

2. Now simplify:

   $$- \sqrt{2} \pi \sin{\left(\pi \left(x + \frac{1}{4}\right) \right)}$$

The answer is:

$$- \sqrt{2} \pi \sin{\left(\pi \left(x + \frac{1}{4}\right) \right)}$$