1. **Problem 135:** Find the derivative of $$y = \frac{2(x^2 + 1)}{\sqrt{\cos 2x}}$$ using logarithmic differentiation. 2. **Step 1:** Take the natural logarithm of both sides: $$\ln y = \ln \left( 2(x^2 + 1) \right) - \ln \left( \sqrt{\cos 2x} \right)$$ 3. **Step 2:** Simplify the logarithms: $$\ln y = \ln 2 + \ln (x^2 + 1) - \frac{1}{2} \ln (\cos 2x)$$ 4. **Step 3:** Differentiate both sides with respect to $x$: $$\frac{1}{y} \frac{dy}{dx} = 0 + \frac{2x}{x^2 + 1} - \frac{1}{2} \cdot \frac{-2 \sin 2x}{\cos 2x}$$ 5. **Step 4:** Simplify the derivative: $$\frac{1}{y} \frac{dy}{dx} = \frac{2x}{x^2 + 1} + \frac{\sin 2x}{\cos 2x} = \frac{2x}{x^2 + 1} + \tan 2x$$ 6. **Step 5:** Multiply both sides by $y$ to solve for $\frac{dy}{dx}$: $$\frac{dy}{dx} = y \left( \frac{2x}{x^2 + 1} + \tan 2x \right) = \frac{2(x^2 + 1)}{\sqrt{\cos 2x}} \left( \frac{2x}{x^2 + 1} + \tan 2x \right)$$ --- 7. **Problem 136:** Find the derivative of $$y = \sqrt{\frac{3x + 4}{2x - 4}}$$ using logarithmic differentiation. 8. **Step 1:** Take the natural logarithm: $$\ln y = \frac{1}{2} \ln \left( \frac{3x + 4}{2x - 4} \right) = \frac{1}{2} \left( \ln (3x + 4) - \ln (2x - 4) \right)$$ 9. **Step 2:** Differentiate both sides: $$\frac{1}{y} \frac{dy}{dx} = \frac{1}{2} \left( \frac{3}{3x + 4} - \frac{2}{2x - 4} \right)$$ 10. **Step 3:** Multiply both sides by $y$: $$\frac{dy}{dx} = y \cdot \frac{1}{2} \left( \frac{3}{3x + 4} - \frac{2}{2x - 4} \right) = \sqrt{\frac{3x + 4}{2x - 4}} \cdot \frac{1}{2} \left( \frac{3}{3x + 4} - \frac{2}{2x - 4} \right)$$ --- 11. **Problem 137:** Find the derivative of $$y = \frac{((t + 1)(t - 1))^5}{(t - 2)(t + 3)}, \quad t > 2$$ using logarithmic differentiation. 12. **Step 1:** Take the natural logarithm: $$\ln y = 5 \ln ((t + 1)(t - 1)) - \ln (t - 2) - \ln (t + 3)$$ 13. **Step 2:** Use log properties: $$\ln y = 5 (\ln (t + 1) + \ln (t - 1)) - \ln (t - 2) - \ln (t + 3)$$ 14. **Step 3:** Differentiate both sides: $$\frac{1}{y} \frac{dy}{dt} = 5 \left( \frac{1}{t + 1} + \frac{1}{t - 1} \right) - \frac{1}{t - 2} - \frac{1}{t + 3}$$ 15. **Step 4:** Multiply both sides by $y$: $$\frac{dy}{dt} = y \left[ 5 \left( \frac{1}{t + 1} + \frac{1}{t - 1} \right) - \frac{1}{t - 2} - \frac{1}{t + 3} \right] = \frac{((t + 1)(t - 1))^5}{(t - 2)(t + 3)} \left[ 5 \left( \frac{1}{t + 1} + \frac{1}{t - 1} \right) - \frac{1}{t - 2} - \frac{1}{t + 3} \right]$$ --- 16. **Problem 138:** Find the derivative of $$y = \frac{2u^2}{\sqrt{u^2 + 1}}$$ using logarithmic differentiation. 17. **Step 1:** Take the natural logarithm: $$\ln y = \ln 2 + 2 \ln u - \frac{1}{2} \ln (u^2 + 1)$$ 18. **Step 2:** Differentiate both sides: $$\frac{1}{y} \frac{dy}{du} = 0 + \frac{2}{u} - \frac{1}{2} \cdot \frac{2u}{u^2 + 1} = \frac{2}{u} - \frac{u}{u^2 + 1}$$ 19. **Step 3:** Multiply both sides by $y$: $$\frac{dy}{du} = y \left( \frac{2}{u} - \frac{u}{u^2 + 1} \right) = \frac{2u^2}{\sqrt{u^2 + 1}} \left( \frac{2}{u} - \frac{u}{u^2 + 1} \right)$$ --- 20. **Problem 139:** Find the derivative of $$y = (\sin \theta)^{\sqrt{\theta}}$$ using logarithmic differentiation. 21. **Step 1:** Take the natural logarithm: $$\ln y = \sqrt{\theta} \ln (\sin \theta)$$ 22. **Step 2:** Differentiate both sides using product rule: $$\frac{1}{y} \frac{dy}{d\theta} = \frac{1}{2 \sqrt{\theta}} \ln (\sin \theta) + \sqrt{\theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{\ln (\sin \theta)}{2 \sqrt{\theta}} + \sqrt{\theta} \cot \theta$$ 23. **Step 3:** Multiply both sides by $y$: $$\frac{dy}{d\theta} = (\sin \theta)^{\sqrt{\theta}} \left( \frac{\ln (\sin \theta)}{2 \sqrt{\theta}} + \sqrt{\theta} \cot \theta \right)$$ --- 24. **Problem 140:** Find the derivative of $$y = (\ln x)^{\frac{1}{\ln x}}$$ using logarithmic differentiation. 25. **Step 1:** Take the natural logarithm: $$\ln y = \frac{1}{\ln x} \ln (\ln x)$$ 26. **Step 2:** Differentiate both sides using quotient and chain rules: $$\frac{1}{y} \frac{dy}{dx} = \frac{(\ln x) \cdot \frac{1}{\ln x} \cdot \frac{1}{x} - \ln (\ln x) \cdot \frac{1}{x}}{(\ln x)^2} = \frac{\frac{1}{x} - \frac{\ln (\ln x)}{x}}{(\ln x)^2} = \frac{1 - \ln (\ln x)}{x (\ln x)^2}$$ 27. **Step 3:** Multiply both sides by $y$: $$\frac{dy}{dx} = (\ln x)^{\frac{1}{\ln x}} \cdot \frac{1 - \ln (\ln x)}{x (\ln x)^2}$$ --- **Summary:** We used logarithmic differentiation to simplify the derivatives of complex functions involving products, quotients, powers, and variable exponents. This method helps by converting products and powers into sums and products inside logarithms, making differentiation easier.