Derivative of x*sqrt(144-x^2)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \sqrt{144 - x^{2}}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 144 - x^{2}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(144 - x^{2}\right)$:

      1. Differentiate $144 - x^{2}$ term by term:

         1. The derivative of the constant $144$ is zero.

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

            1. Apply the power rule: $x^{2}$ goes to $2 x$

            So, the result is: $- 2 x$

         The result is: $- 2 x$

      The result of the chain rule is:

      $$- \frac{x}{\sqrt{144 - x^{2}}}$$

   The result is: $- \frac{x^{2}}{\sqrt{144 - x^{2}}} + \sqrt{144 - x^{2}}$

2. Now simplify:

   $$\frac{2 \left(72 - x^{2}\right)}{\sqrt{144 - x^{2}}}$$

The answer is:

$$\frac{2 \left(72 - x^{2}\right)}{\sqrt{144 - x^{2}}}$$