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

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

Discussion

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