Derivative of cos^25x-sin^25x

1. Differentiate $- \sin^{25}{\left(x \right)} + \cos^{25}{\left(x \right)}$ term by term:

   1. Let $u = \cos{\left(x \right)}$.

   2. Apply the power rule: $u^{25}$ goes to $25 u^{24}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$:

      1. The derivative of cosine is negative sine:

         $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

      The result of the chain rule is:

      $$- 25 \sin{\left(x \right)} \cos^{24}{\left(x \right)}$$

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

      1. Let $u = \sin{\left(x \right)}$.

      2. Apply the power rule: $u^{25}$ goes to $25 u^{24}$

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

         1. The derivative of sine is cosine:

            $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

         The result of the chain rule is:

         $$25 \sin^{24}{\left(x \right)} \cos{\left(x \right)}$$

      So, the result is: $- 25 \sin^{24}{\left(x \right)} \cos{\left(x \right)}$

   The result is: $- 25 \sin^{24}{\left(x \right)} \cos{\left(x \right)} - 25 \sin{\left(x \right)} \cos^{24}{\left(x \right)}$

2. Now simplify:

   $$- 25 \left(\sin^{23}{\left(x \right)} + \cos^{23}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$

The answer is:

$$- 25 \left(\sin^{23}{\left(x \right)} + \cos^{23}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$