Questions: Using logarithmic differentiation, find the derivative of y=(x^9 sqrt[5]x^2+13)/(x^3+27) dy/dx=9/x+2x/5(x^2+13)+3x^2/(x^3+27) dy/dx=(x^9 sqrt[5]x^2+13)/(x^3+27)(9/x+2x/5(x^2+13)+3x^2/(x^3+27)) dy/dx=9/x+2x/5(x^2+13)-3x^2/(x^3+27) dy/dx=(x^9 sqrt[5]x^2+13)/(x^3+27)(9/x+2x/5(x^2+13)-3x^2/(x^3+27))

Find the derivative of \( y = \frac{x^9 \sqrt[5]{x^2 + 13}}{x^3 + 27} \) using logarithmic differentiation.

Take the natural logarithm of both sides.

We have \( \ln(y) = \ln\left(\frac{x^9 \sqrt[5]{x^2 + 13}}{x^3 + 27}\right) = \ln(x^9) + \ln\left(\sqrt[5]{x^2 + 13}\right) - \ln(x^3 + 27) \).

Differentiate both sides with respect to \( x \).

Differentiating gives us \( \frac{1}{y} \frac{dy}{dx} = \frac{9}{x} + \frac{2x}{5(x^2 + 13)} - \frac{3x^2}{x^3 + 27} \).

Solve for \( \frac{dy}{dx} \).

Multiplying both sides by \( y \) results in \( \frac{dy}{dx} = y \left( \frac{9}{x} + \frac{2x}{5(x^2 + 13)} - \frac{3x^2}{x^3 + 27} \right) \).

Substituting back for \( y \).

Thus, we have \( \frac{dy}{dx} = \frac{x^9 \sqrt[5]{x^2 + 13}}{x^3 + 27} \left( \frac{9}{x} + \frac{2x}{5(x^2 + 13)} - \frac{3x^2}{x^3 + 27} \right) \).

\(\boxed{\frac{dy}{dx} = \frac{x^9 \sqrt[5]{x^2 + 13}}{x^3 + 27} \left( \frac{9}{x} + \frac{2x}{5(x^2 + 13)} - \frac{3x^2}{x^3 + 27} \right)}\)

The derivative is given by \( \frac{dy}{dx} = \frac{x^9 \sqrt[5]{x^2 + 13}}{x^3 + 27} \left( \frac{9}{x} + \frac{2x}{5(x^2 + 13)} - \frac{3x^2}{x^3 + 27} \right) \).