Differentiate each of the following from first principle: `tansqrt(x)`

Explanation

`text { By using first principle, } f^{prime}(x)=lim _{h -> 0} frac{f(x+h)-f(x)}{h}`


`=lim _{h -> 0} frac{tan sqrt{x+h}-tan sqrt{x}}{h} `


=lim _{h -> 0} frac{frac{sin sqrt{x+h}}{cos sqrt{x+h}} frac{sin sqrt{x}}{cos sqrt{x}}}{h}`


=lim _{h -> 0} frac{sin sqrt{x+h} cos sqrt{x}-sin sqrt{x} cos sqrt{x+h}}{h cos (sqrt{x+h}) cos sqrt{x}}`


`=lim _{h -> 0} frac{sin [(sqrt{x+h})-sqrt{x}]}{cos sqrt{x} cos (sqrt{(x+h}))} quad[because sin A cos B-cos A sin B=sin (A-B)]`


`=lim _{h -> 0} frac{sin (sqrt{x+h}-sqrt{x})}{h cos sqrt{x} cos (sqrt{x+h})} times frac{sqrt{x+h}-sqrt{x}}{sqrt{x+h}-sqrt{x}}`


`text { [multipying numerator and denominator by } sqrt{x+h}-sqrt{x} text { ] }`


`=lim _{h -> 0} frac{sqrt{x+h}-sqrt{x} x}{h cos sqrt{x} cos (sqrt{x+h})} times lim _{h -> 0}(frac{sin (sqrt{x+h}-sqrt{x})}{sqrt{x+h}-sqrt{x}})`


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