How do you differentiate $f (x) = \frac{1}{\sqrt{x - 3}}$ $f(x)=1/(sqrt(x-3)$ using the chain rule?

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How do you differentiate $f (x) = \frac{1}{\sqrt{x - 3}}$ $f(x)=1/(sqrt(x-3)$ using the chain rule?

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Answer 1 · 2016-01-01T18:09:36

Chain rule states that $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)(du)/(dx)$ .
We'll also need two power rules here.

Explanation:

$a^{- n} = \frac{1}{a^{n}}$ $a^-n=1/a^n$
$a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ $a^(m/n)=root(n)(a^m)$

So, we can rewrite it as $f (x) = {(x - 3)}^{- \frac{1}{2}}$ $f(x)=(x-3)^(-1/2)$

Renaming $u = x - 3$ $u=x-3$ , we get $f (x) = u^{- \frac{1}{2}}$ $f(x)=u^(-1/2)$ and can now differentiate it.

$\frac{d y}{d x} = - \frac{1}{2} u^{- \frac{3}{2}} (1) = - \frac{1}{2 (u^{\frac{3}{2}})} = - \frac{1}{2 \sqrt{{(x - 3)}^{3}}}$ $(dy)/(dx)=-1/2u^(-3/2)(1)=-1/(2(u^(3/2)))=-1/(2sqrt((x-3)^3))$

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How do you differentiate $f (x) = \frac{1}{\sqrt{x - 3}}$ $f(x)=1/(sqrt(x-3)$ using the chain rule?

1 Answer

Explanation:

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How do you differentiate f(x)=1√x−3f(x)=1/(sqrt(x-3) using the chain rule?

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How do you differentiate $f (x) = \frac{1}{\sqrt{x - 3}}$ $f(x)=1/(sqrt(x-3)$ using the chain rule?