# how to differentiate a function with square root

Trying to solve

$y =7t^4-10 \sqrt {t+\frac{10}{t}}$

I know how to differentiate down to $7(4t^3)- . . .$ and I know a sqrt is equal to $x^.5$ but cannot figure out how to apply that to the rest of the function. Please help!

thanks

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Do you know the chain rule? –  Daniel Littlewood Feb 27 at 22:43

Here's a hint:

$$\frac{d}{dx}\sqrt{x} = \frac{d}{dx} x^{\frac{1}{2}}.$$

By power rule, we know that $\frac{d}{dx} x^{\alpha} = \alpha x^{\alpha-1}$. Applying this for the square root, we have

$$\frac{d}{dx}\sqrt{x} = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}}.$$

Then apply the chain rule and you will get the answer.

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differentiate the function $\sqrt{f(x)}$ with respect to $f(x)$ just like algebraic function taking exponent $\frac{1}{2}$ and multiply it with the differentiation of $f(x)$ with respect to $x$

$\frac{d}{dx}\sqrt{f(x)}=\frac{1}{2\sqrt{f(x)}}\frac{d}{dx}f(x)$

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