Derivative of exp(2x-1)-sin(2x-1)

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

   1. Let $u = 2 x - 1$.

   2. The derivative of $e^{u}$ is itself.

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 1\right)$:

      1. Differentiate $2 x - 1$ term by term:

         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: $2$

         2. The derivative of the constant $-1$ is zero.

         The result is: $2$

      The result of the chain rule is:

      $$2 e^{2 x - 1}$$

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

      1. Let $u = 2 x - 1$.

      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 $\frac{d}{d x} \left(2 x - 1\right)$:

         1. Differentiate $2 x - 1$ term by term:

            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: $2$

            2. The derivative of the constant $-1$ is zero.

            The result is: $2$

         The result of the chain rule is:

         $$2 \cos{\left(2 x - 1 \right)}$$

      So, the result is: $- 2 \cos{\left(2 x - 1 \right)}$

   The result is: $2 e^{2 x - 1} - 2 \cos{\left(2 x - 1 \right)}$

2. Now simplify:

   $$2 e^{2 x - 1} - 2 \cos{\left(2 x - 1 \right)}$$

The answer is:

$$2 e^{2 x - 1} - 2 \cos{\left(2 x - 1 \right)}$$