Derivative of 3/x-3-2sin(x-1)

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

   1. Differentiate $-3 + \frac{3}{x}$ 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: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$

         So, the result is: $- \frac{3}{x^{2}}$

      2. The derivative of the constant $-3$ is zero.

      The result is: $- \frac{3}{x^{2}}$

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

      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:

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

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

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

2. Now simplify:

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

The answer is: