Differentiate $\frac{v^4-10v^2\sqrt{v}}{4v^2}$?

Question

I am trying to differentiate $\frac{v^4-10v^2\sqrt{v}}{4v^2}$.

I have tried splitting the fraction and doing the division before finding the differential, but I am still not getting the right answer.

$$ \frac{v^4-10v^2\sqrt{v}}{4v^2} =\frac{v^4}{4v^2}-\frac{10v^2\sqrt{v}}{4v^2} = \frac{v^2}{4}-\frac{5\sqrt{v}}{2} $$

To find the differentials:

$$\frac{d}{dv}v^2=2\cdot v^{2-1}=2v$$

$$\frac{d}{dv} 5\sqrt{v}=5 \cdot \frac{d}{dv}\sqrt{v}=5 \cdot \frac{1}{2}v^{\frac{1}{2}- \frac{2}{2}}=5 \frac{\frac{1}{\sqrt{v}}}{2}$$

So I got:

$$2v - 5 \frac{\frac{1}{\sqrt{v}}}{2}$$

but this is wrong.

Answer 1

You are forgetting to divide the first term by $4$, and the second term by $2$.

Answer 2

Hint: To make your work a little easier...

Let $w = \sqrt {v}$. Then $w^2 = v$, $w^4 = v^2$, and $w^8 = v^4.$ You have $$\dfrac {w^8 - 10w^5}{4w^4}$$ and then simplifying you can get $$\dfrac {1}{4} w^4 - \dfrac {10}{4} w$$ Resubsitute $w = \sqrt {v}$ and take it from there.

I got the answer $$\dfrac {2v-5\sqrt{v}}{4v}\Leftrightarrow \dfrac {v}{2} - \dfrac {5 \sqrt {v}}{4v}$$

Answer 3

After simplifying the expression from...

$$ \frac{v^4-10v^2\sqrt{v}}{4v^2} $$.

To...

$$\frac{v^2-10\sqrt{v}}{4}$$.

I evaluated and had gotten...

$$\dfrac {v}{2} - \dfrac {5}{\sqrt {v}}$$