Derivative of sin(x²)-cos(x)

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

   1. Let $u = x^{2}$.

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

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

      The result of the chain rule is:

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

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

      1. The derivative of cosine is negative sine:

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

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

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

The answer is:

$$2 x \cos{\left(x^{2} \right)} + \sin{\left(x \right)}$$