Derivative of 4/(x-3)-16/(x-7)

1. Differentiate $\frac{4}{x - 3} - \frac{16}{x - 7}$ term by term:

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

      1. Let $u = x - 3$.

      2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

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

         1. Differentiate $x - 3$ term by term:

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

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

            The result is: $1$

         The result of the chain rule is:

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

      So, the result is: $- \frac{4}{\left(x - 3\right)^{2}}$

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

      1. Let $u = x - 7$.

      2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x - 7\right)$:

         1. Differentiate $x - 7$ term by term:

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

            2. The derivative of the constant $-7$ is zero.

            The result is: $1$

         The result of the chain rule is:

         $$- \frac{1}{\left(x - 7\right)^{2}}$$

      So, the result is: $\frac{16}{\left(x - 7\right)^{2}}$

   The result is: $- \frac{4}{\left(x - 3\right)^{2}} + \frac{16}{\left(x - 7\right)^{2}}$

2. Now simplify:

   $$- \frac{4}{\left(x - 3\right)^{2}} + \frac{16}{\left(x - 7\right)^{2}}$$

The answer is:

$$- \frac{4}{\left(x - 3\right)^{2}} + \frac{16}{\left(x - 7\right)^{2}}$$