Derivative of sin(x-1)+ln((2x+3)/5)

1. Differentiate $\log{\left(\frac{2 x + 3}{5} \right)} + \sin{\left(x - 1 \right)}$ term by term:

   1. Let $u = x - 1$.

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

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

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

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

         The result is: $1$

      The result of the chain rule is:

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

   4. Let $u = \frac{2 x + 3}{5}$.

   5. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

   6. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 3}{5}$:

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

         1. Differentiate $2 x + 3$ 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 $3$ is zero.

            The result is: $2$

         So, the result is: $\frac{2}{5}$

      The result of the chain rule is:

      $$\frac{2}{2 x + 3}$$

   The result is: $\cos{\left(x - 1 \right)} + \frac{2}{2 x + 3}$

2. Now simplify:

   $$\frac{\left(2 x + 3\right) \cos{\left(x - 1 \right)} + 2}{2 x + 3}$$

The answer is:

$$\frac{\left(2 x + 3\right) \cos{\left(x - 1 \right)} + 2}{2 x + 3}$$