Question

give the derivative of √a+√x/√a-√x

Answer 1

answer of this is here by picture

Answer 2

The first derivative is ${\dfrac{(a-x)(1+\sqrt{a}+\sqrt{a}\dfrac{1}{\sqrt{x}})-(a+x+2\sqrt{a}\sqrt{x})(-1)}{(a-x)^{2}}}$

Step-by-step explanation:

Let $y=\dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}$

To find, derivative of $\dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}=?$

∴ $y=\dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}\times \dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a}+\sqrt{x}}$

$=\dfrac{(\sqrt{a}+\sqrt{x})^2}{\sqrt{a}^2-\sqrt{x}^2}=\dfrac{a+x+2\sqrt{a}\sqrt{x}}{a-x}$

We know that,

$\dfrac{d(\dfrac{u}{v} )}{dx} =\dfrac{v\dfrac{du}{dx}-u\dfrac{dv}{dx} }{v^{2}}$

$={\dfrac{(a-x)(1+\sqrt{a}+2\sqrt{a}\dfrac{1}{2\sqrt{x}})-(a+x+2\sqrt{a}\sqrt{x})(-1)}{(a-x)^{2}}}$

Hence, the first derivative is ${\dfrac{(a-x)(1+\sqrt{a}+\sqrt{a}\dfrac{1}{\sqrt{x}})-(a+x+2\sqrt{a}\sqrt{x})(-1)}{(a-x)^{2}}}$