Derivative of e^x*sinx-1

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

   1. Apply the product rule:

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

      $f{\left(x \right)} = e^{x}$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. The derivative of $e^{x}$ is itself.

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

      1. The derivative of sine is cosine:

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

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

   2. The derivative of the constant $\left(-1\right) 1$ is zero.

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

2. Now simplify:

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

The answer is: