1. \[ y’ = x^{x^3+2}(1+3\ln x) \]
2. \[ y’ = \frac{1}{2}x^{\sqrt{x}-\frac{1}{2}}(2+\ln x) \]
3. \[ y’ = 2\ln x \cdot x^{\ln x-1} \]
4. \[ y’ = \frac{1}{x}(\ln x)^{\ln x}(1+\ln(\ln x)) \]
5. \[ y’ = (\ln 2)(\ln 3)3^x 2^{3^x} \]
6. \[ y’ = a^x x^a \left(\frac{a}{x} + \ln a\right) \]
7. \[ y’ = \sqrt[x]{x} \left( \frac{1-\ln x}{x^2} \right) \]
8. \[ y’ = (x^2+1)^{\operatorname{sen} x} \left( \frac{2x \operatorname{sen} x}{x^2+1} + \cos x \ln (x^2+1) \right) \]
9. \[ y’ = (\operatorname{sen} x)^{\cos x} \left( \frac{\cos^2 x}{\operatorname{sen} x} – \operatorname{sen} x \ln(\operatorname{sen} x) \right) \]
10. \[ y’ = \left(1+\frac{1}{x}\right)^x \left( \ln \frac{x+1}{x} – \frac{1}{x+1} \right) \]
11. \[ y’ = \frac{x(x^2-1)}{\sqrt{x^2+1}} \left( \frac{1}{x} + \frac{2x}{x^2-1} – \frac{x}{x^2+1} \right) \]
12. \[ y’ = \frac{1}{3}\sqrt[3]{\frac{x(x^2-1)}{(x+1)^2}} \left( \frac{1}{x} + \frac{2x}{x^2-1} – \frac{2}{x+1} \right) \]

Hallar la derivada de las siguientes funciones empleando derivación logarítmica:

1. \[ y = x^{x^3} \]
2. \[ y = x^{\sqrt{x}}, \, x > 0 \]
3. \[ y = x^{\ln x}, \, x > 0 \]
4. \[ y = (\ln x)^{\ln x} \]
5. \[ y = 2^{3^x} \]
6. \[ y = a^x x^a \]
7. \[ y = \sqrt[x]{x} \]
8. \[ y = (x^2 + 1)^{\operatorname{sen} x} \]
9. \[ y = (\operatorname{sen} x)^{\cos x} \]
10. \[ y = \left( 1 + \frac{1}{x} \right)^x \]
11. \[ y = \frac{x (x^2 – 1)}{\sqrt{x^2 + 1}} \]
12. \[ y = \sqrt[3]{\frac{x (x^2 – 1)}{(x + 1)^2}} \]