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Find the derivative of sqrt(tanx) using ...

Find the derivative of `sqrt(tanx)` using first principles.

Text Solution

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`f(x) = sqrt(tanx)`
`Lim_(h->0) (f(x+h)-f(x))/h = Lim_(h->0) (sqrt(tan(x+h))-sqrt(tanx))/h`
Multiplying nunerator and denominator by `(sqrt(tan(x+h))+sqrt(tanx))`,
` =Lim_(h->0) (tan(x+h) - (tanx))/ ((sqrt(tan(x+h))+sqrt(tanx))h)`
`=Lim_(h->0) ((sin h)/(cos(x+h)cosx))/ ((sqrt(tan(x+h))+sqrt(tanx))h)`
`= Lim_(h->0) sin h/h xx Lim_(h->0)(1/((cos(x+h)cosx)(sqrt(tan(x+h))+sqrt(tanx))))`
`=1 xx 1/(2cos^2xsqrt(tanx))`
`=sec^2x/(2sqrttan x)`
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