it is impossible for me to find derivative of the function f(x)=x^(2*square root x)

giorgiana1976 | Student

To find derivative of the function, first we'll take natural logarithms both sides:

ln f(x) = ln x^(2*sqrt x)

We'll use the property of logarithms:

ln f(x) = (2*sqrt x)*ln x

Now, we'll differentiate both sides, with respect to x:

[ln f(x)]' = [(2*sqrt x)*ln x]'

We'll apply the product rule to the right side:

f'(x)/f(x) = 2*ln x/2sqrtx + 2*sqrt x/x

f'(x)/f(x) = ln x/sqrtx + 2*sqrt x/x

f'(x)/f(x) = (sqrt x*ln x + 2sqrt x)/x

f'(x) = f(x)*(sqrt x*ln x + 2sqrt x)/x

f'(x) = [x^(2sqrt x)]*(sqrt x)*(ln x + 2)/x

The first derivative of the given function is: f'(x) = [x^(2sqrt x)]*(sqrt x)*(ln x + 2)/x

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