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Given:
$a=\frac{1}{2} y^{3}+5 y+2$
Determine $\frac{d}{d y} a$ using the rules for differentiation.
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Recall the rule for differentiation
The rule for differentiation tells us to multiply the coefficient of each term by the exponent, and then subtract one from the exponent.
The rule for differentiation is:
$\frac{d}{d x} a x^{n}=(a n) x^{n-1}$

Differentiate the first term
Let's consider the first term of the function:
\begin{aligned} \frac{d}{d y}\left(\frac{1}{2} y^{3}\right) &=3\left(\frac{1}{2}\right) y^{(3-1)} \\ &=\frac{3}{2} y^{2} \end{aligned}
We follow the same rule for all the remaining terms.
Differentiate the rest of the function
\begin{aligned} &\frac{d}{d y}\left(\frac{1}{2} y^{3}+5 y+2\right) \\ &=3\left(\frac{1}{2}\right) y^{(3-1)}+1(5) y^{(1-1)}+0(2) y^{(0-1)} \\ &=\frac{3 y^{2}}{2}+5 \end{aligned}
Therefore, we can write the final answer:
$\frac{d}{d y} a=\frac{3 y^{2}}{2}+5$
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