Question: "Derivatives of products and quotients Find the derivative..."

lugreget9

Answered question

2021-12-11

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression.

y = \frac{x - a}{\sqrt{x} - \sqrt{a}}

; a is a positive ocnstant.

twineg4

Beginner2021-12-12Added 33 answers

Step 1
Given:

y = \frac{x - a}{\sqrt{x} - \sqrt{a}}

, where a is constant
Step 2
Simplify the given equation

y = \frac{x - a}{\sqrt{x} - \sqrt{a}}

= \frac{{(\sqrt{x})}^{2} - {(\sqrt{a})}^{2}}{\sqrt{x} - \sqrt{a}}

= \frac{(\sqrt{x} - \sqrt{a}) (\sqrt{x} + \sqrt{a})}{(\sqrt{x} - \sqrt{a})}

... Use

m^{2} - n^{2} = (m - n) (m + n)

= \sqrt{x} + \sqrt{a}

Thus,

y = \sqrt{x} + \sqrt{a}

Differentiate the above simplified equation with respect to x, to get

\frac{d y}{d x} = \frac{d}{d x} (x^{\frac{1}{2}}) + \frac{d}{d x} (a^{\frac{1}{2}})

= \frac{1}{2} x^{\frac{1}{2} - 1} + 0

... Use

\frac{d}{d x} (x^{n}) = n x^{n - 1} and \frac{d}{d x} (c) = 0

, where c is constant

= \frac{1}{2} x^{(- \frac{1}{2})}

= \frac{1}{2 x^{(\frac{1}{2})}}

= \frac{1}{2 \sqrt{x}}

Step 3
Thus, the derivative of

y = \frac{x - a}{\sqrt{x} - \sqrt{a}}

, where a is constant is:

\frac{1}{2 \sqrt{x}}

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