Derivative Quotient 58478B

1. **State the problem:** We need to find the derivative of the function $$f(x) = \frac{\sqrt{x} - 5}{\sqrt{x} + 5}$$. 2. **Recall the formula:** For a function $$f(x) = \frac{u(x)}{v(x)}$$, the derivative is given by the quotient rule: $$ f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} $$ 3. **Identify components:** Here, $$ u(x) = \sqrt{x} - 5 = x^{\frac{1}{2}} - 5$$ $$v(x) = \sqrt{x} + 5 = x^{\frac{1}{2}} + 5$$ 4. **Find derivatives of numerator and denominator:** $$u'(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$$ $$v'(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$$ 5. **Apply the quotient rule:** $$ f'(x) = \frac{(\sqrt{x} + 5)\left(\frac{1}{2\sqrt{x}}\right) - (\sqrt{x} - 5)\left(\frac{1}{2\sqrt{x}}\right)}{(\sqrt{x} + 5)^2} $$ 6. **Factor out common terms in numerator:** $$ f'(x) = \frac{\frac{1}{2\sqrt{x}}\left[(\sqrt{x} + 5) - (\sqrt{x} - 5)\right]}{(\sqrt{x} + 5)^2} $$ 7. **Simplify inside the brackets:** $$ (\sqrt{x} + 5) - (\sqrt{x} - 5) = \sqrt{x} + 5 - \sqrt{x} + 5 = 10 $$ 8. **Substitute back:** $$ f'(x) = \frac{\frac{1}{2\sqrt{x}} \times 10}{(\sqrt{x} + 5)^2} = \frac{10}{2\sqrt{x}(\sqrt{x} + 5)^2} $$ 9. **Simplify the fraction:** $$ f'(x) = \frac{5}{\sqrt{x}(\sqrt{x} + 5)^2} $$ **Final answer:** $$ f'(x) = \frac{5}{\sqrt{x}(\sqrt{x} + 5)^2} $$