Derivative of (sin(t)-cos(t))*t

1. Apply the product rule:

   $$\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}$$

   $f{\left(t \right)} = \sin{\left(t \right)} - \cos{\left(t \right)}$; to find $\frac{d}{d t} f{\left(t \right)}$:

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

      1. The derivative of sine is cosine:

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

      2. 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:

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

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

      The result is: $\sin{\left(t \right)} + \cos{\left(t \right)}$

   $g{\left(t \right)} = t$; to find $\frac{d}{d t} g{\left(t \right)}$:

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

   The result is: $t \left(\sin{\left(t \right)} + \cos{\left(t \right)}\right) + \sin{\left(t \right)} - \cos{\left(t \right)}$

2. Now simplify:

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

The answer is:

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