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Find the derivative of e^(sqrt(x)) w.r.t...

Find the derivative of `e^(sqrt(x))` w.r.t. `x` using the first principle.

Text Solution

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Let `f(x)=""_(e)sqrt(x)`. Then `f(x+h)=""_(e)sqrt(x+h)`
`therefore" "(d)/(dx)(f(x))=underset(hrarr0)lim(f(x+h)-f(x))/(h)`
`=underset(hrarr0)lim(""_(e)sqrt(x+h)-""_(e)sqrt(x))/(h)`
`=""_(e)sqrt(x)" "underset(hrarr0)lim(""_(e)sqrt(x+h)-sqrt(x)-1)/(h)`
`=""_(e)sqrt(x)underset(hrarr0)lim((esqrt(x+h)-sqrt(x)-1)/(sqrt(x+h)-sqrt(x)))((sqrt(x+h)-sqrt(x))/(h))`
`=""_(e)sqrt(x)underset(hrarr0)lim((esqrt(x+h)-sqrt(x)-1)/(sqrt(x+h)-sqrt(x)))`
`x((sqrt(x+h)-sqrt(x))(sqrt(x+h)+sqrt(x)))/(h(sqrt(x+h)+sqrt(x)))`
`=""_(e)sqrt(x)underset(hrarr0)lim((e^(y)-1)/(y))underset(hrarr0)lim((x+h-x)/(h(sqrt(x+h)+sqrt(x))))`
`"where "y=sqrt(x+h)-sqrt(x)" "(because" when " hrarr0,yrarr0)`
`therefore" "(d)/(dx)(f(x))=""_(e)sqrt(x)xx1xx((1)/(sqrt(x)+sqrt(x)))=(""_(e)sqrt(x))/(2sqrt(x))`
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