Question

differentiate it w.r.t x​

Answer 1

Question:-

Differenitaiate:-{(a+x)^1/2 -(a-x)^1/2}/{(a+x)^1/2 +(a-x)^1/2}

solution:-

First rationalize the term....and make it simple..

multiplying numerator and denominator by (a+x)^1/2 -(a-x)^1/2

we get

{(a+x)^1/2 -(a-x)^1/2}{(a+x)^1/2 -(a-x)^1/2}/{(a+x)^1/2 +(a-x)^1/2}{(a+x)^1/2 -(a-x)^1/2}

{(a+x)^1/2 -(a-x)^1/2}^2/{(a+x-a+x)

{(a+x+a-x-2×(a+x)^1/2 ×(a-x)^1/2}/2x

{(2a-2(a+x)^1/2 ×(a-x)^1/2}/2x

{a-(a+x)^1/2×(a-x)^1/2}/x

{a-{a^2-x^2}^1/2}/x

let,y={a-(a^2-x^2)^1/2}/x......i)

now,use the quotient rule of differenitaiation

dy/dx=[{0-(0-2x)×1/2 (a^2-x^2)^-1/2 - 1×{a-(a^2-x^2)^1/2}]/x^2

=>[{x^2/(a^2-x^2)^-1/2} -{a-(a^2-x^2)^1/2}]/x^2

=>[x^2-a(a^2-x^2)^1/2 +a^2-x^2}]/x^2(a^2-x^2)^1/2

=>[a^2-a(a^2-x^2)^1/2]/(x^2(a^2-x^2)^1/2)

=>[a{a-(a^2-x^2)^1/2}/{x^2(a^2-x^2)^1/2}]

=>from first equation put y

=>a.y/{x(a^2-x^2)^1/2)

so,

dy/dx =a×y/(x(a^2-x^2)^1/2)

{hope it helps}

Answer 2

Answer:

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