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QUESTION: differentiate w.r.t x :e^1/2*x^1/2
ANSWER: The 1st term, e^(1/2), is √e, and that's a constant.
The 2nd term, x^(1/2), has a derivative following the derivative of x^n is nx^(n-1).
Since, in this case, n = 1/2, n-1 = -1/2, that gives the derivative.
It is √e(1/2)x^(-1/2). This can be rewritten as √e/(2√x).
Since denominators are frequently rationalized, that is √(ex)/2x.
---------- FOLLOW-UP ----------
QUESTION: here 'e' is not a constant.it is the natural exponential function.please reconsider your solution
In the exponential function f(x) = e^x, e is a constant and is defined as e = 2.7182818245905...
as seen in http://en.wikipedia.org/wiki/Exponential_function
The exponet x makes the function vary, but e is still a constant.
It is a constant just like in trig where pi is a constant defined as pi = 3.14159285358979323...
Both e and pi are non-repeating decimals with an infinite number of places past the decimal.
One of the uses of the number e is that if f(x) = e^x, then f'(x) = e^x.
Here, we have what looks like [e^(1/2)][x^(1/2)] = (e^0.5)*(x^0.5) = √e√x.
That, √e√x, is the same as saying (1.648721271)√x, since e^0.5 = 1.648721271...
As ws stated, the derivative would be √e/(2√x),
which rationalizes to √(ex)/(2x).
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