
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Find the derivative of the following functions:

(1) $y=(x+1)(x^{2}-1)$;

(2) $y=\frac{x-1}{x+1}$;

(3) $y=\sin 2x$;

(4) $y=e^{x}\cdot \ln x$;

(5) $y=\log_{2}(2x-1)+\frac{\sin x}{e^{x}}$;

(6) $y=3^{x}\cdot \cos \left( \frac{1}{2}x-\frac{\pi}{6} \right)+x\cdot \ln x$.

Present the answer in LaTex format: \boxed{Your answer}
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