Week 7 Worksheet

Differentiate the following functions:

1. \(y = -5e^{3x+2}\)
2. \(y = 4e^{2x^2-4}\)
3. \(y = \frac{x^2}{e^x}\)
4. \(y = 4^{-5x + 2}\)
5. \(y = 3 \cdot 4^{x^2+2}\)
6. \(y = \frac{x^2e^{2x}}{x+e^{3x}}\)
7. \(y = \ln(4x)\)
8. \(y = \ln \lvert 4x^2 - 9x \rvert\)
9. \(y = \ln \sqrt{x + 6}\)
10. \(y = \ln \lvert \ln x \rvert\)
11. \(y = \log_3 (x^2 + 2x)^{3/2}\)
12. \(y = \frac{\ln(t^2+1) + t}{\ln(t^2+1)+1}\)
13. \(y = (x^2+1)^{5x}\)
14. \(y = \log_2(x^2 + x + 1)\)
15. \(y = \frac{2x^{3/2}}{\ln (2x^{3/2} + 1)}\)
16. \(y = x^{\ln x}\)
17. (Challenge) Prove that the only positive integers \(a, b\) such that \(a^b = b^a\) and \(a \neq b\) are \(2\) and \(4\). (Hint: Re-write so it looks like \(f(a) = f(b)\), then think about when \(f\) is increasing and decreasing.)