Differentiate the following equations: a) y = (ln x)/x2 b) y = (x3 + 3x)5 c) y = e^(2x) sin x d) y = 2x4 − 5x2 + 6x + 1

Let $u = \ln x \implies u' = \frac{1}{x}$.
Let $v = x^2 \implies v' = 2x$.

Let $g(x) = x^3 + 3x$, so $g'(x) = 3x^2 + 3$.
Let $f(u) = u^5$, so $f'(u) = 5u^4$.

Let $u = e^{2x} \implies u' = 2e^{2x}$.
Let $v = \sin x \implies v' = \cos x$.

For $n=1: U_1 = 2(1) + 1 = 3$
For $n=2: U_2 = 2(2) + 1 = 5$
For $n=3: U_3 = 2(3) + 1 = 7$
For $n=4: U_4 = 2(4) + 1 = 9$
The first four terms are $\boxed{3, 5, 7, 9}$.

II. Hence, write down the first four terms of the sequence ($V_n$).

For $n=1: V_1 = U_1^2 - 3 = 3^2 - 3 = 9 - 3 = 6$
For $n=2: V_2 = U_2^2 - 3 = 5^2 - 3 = 25 - 3 = 22$
For $n=3: V_3 = U_3^2 - 3 = 7^2 - 3 = 49 - 3 = 46$
For $n=4: V_4 = U_4^2 - 3 = 9^2 - 3 = 81 - 3 = 78$
The first four terms are $\boxed{6, 22, 46, 78}$.

III. Show that $V_n = 4n^2 + 4n - 2$.

IV. Determine whether the sequence $U_n$ and $V_n$ is a GP or AP.

The terms of $U_n$ are $3, 5, 7, 9, \dots$
Common difference: $U_2 - U_1 = 5 - 3 = 2$. $U_3 - U_2 = 7 - 5 = 2$.
Since there is a constant common difference, $U_n$ is an Arithmetic Progression (AP).
Common ratio: $\frac{U_2}{U_1} = \frac{5}{3}$. $\frac{U_3}{U_2} = \frac{7}{5}$. Since the ratios are not equal, $U_n$ is not a GP.

The terms of $V_n$ are $6, 22, 46, 78, \dots$
Common difference: $V_2 - V_1 = 22 - 6 = 16$. $V_3