use logarithmic differentiation to find the derivative of the function.\n\n$y = \\sqrt{\\frac{x - 3}{x^{8} +…

Explanation:

Step1: Take natural logarithm on both sides

$$\ln y=\ln\sqrt{\frac{x - 3}{x^{8}+4}}=\frac{1}{2}\ln\frac{x - 3}{x^{8}+4}=\frac{1}{2}(\ln(x - 3)-\ln(x^{8}+4))$$

Step2: Differentiate both sides with respect to (x)

Using the chain rule ((\ln u)^\prime=\frac{u^\prime}{u}), we have: $$\frac{y^\prime}{y}=\frac{1}{2}(\frac{1}{x - 3}-\frac{8x^{7}}{x^{8}+4})$$

Step3: Solve for (y^\prime)

Multiply both sides by (y=\sqrt{\frac{x - 3}{x^{8}+4}}): $$y^\prime=\sqrt{\frac{x - 3}{x^{8}+4}}\times\frac{1}{2}(\frac{1}{x - 3}-\frac{8x^{7}}{x^{8}+4})$$ Simplify the expression: $$y^\prime=\frac{1}{2}\sqrt{\frac{x - 3}{x^{8}+4}}\left(\frac{x^{8}+4-8x^{7}(x - 3)}{(x - 3)(x^{8}+4)}\right)$$ $$y^\prime=\frac{x^{8}+4-8x^{8}+24x^{7}}{2(x - 3)^{\frac{1}{2}}(x^{8}+4)^{\frac{3}{2}}}$$ $$y^\prime=\frac{-7x^{8}+24x^{7}+4}{2(x - 3)^{\frac{1}{2}}(x^{8}+4)^{\frac{3}{2}}}$$

Answer:

(y^\prime=\frac{-7x^{8}+24x^{7}+4}{2\sqrt{x - 3}(x^{8}+4)^{\frac{3}{2}}})