## Related questions with answers

Use the General Power Rule to find the derivative of the function. $s(t)=\sqrt{2 t^{2}+5 t+2}$

Solutions

VerifiedObjective is to find the derivative of the function by general power rule,

$s(t)=\sqrt{2t^2+5t+2}$

We need to find the derivative of the given function

$\begin{equation} s(t)=\sqrt{2 t^{2}+5 t+2} \end{equation}$

In this case, the respective inner and outer functions are

$\begin{equation} u=2 t^{2}+5 t+2, \quad s(u)=\sqrt{u} \end{equation}$

then

$\begin{equation} \begin{aligned} \frac{d s}{d t} &=\frac{d}{d t}\left(u^{1 / 2}\right)=\frac{1}{2 \sqrt{u}} \frac{d u}{d t} \quad(\text { general power rule }) \\ &=\frac{1}{2 \sqrt{u}} \frac{d}{d t}\left(2 t^{2}+5 t+2\right) \\ &=\frac{1}{2 \sqrt{u}}(4 t+5) \\ &=\frac{4 t+5}{2 \sqrt{2 t^{2}+5 t+2}} \quad \text { substitute } u=2 t^{2}+5 t+2 \end{aligned} \end{equation}$

Finally, we have found that

$\begin{equation} \frac{d s}{d t}=\frac{4 t+5}{2 \sqrt{2 t^{2}+5 t+2}} \end{equation}$

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