Find f ' (1) if f(x)=n√x
or x^1/n
options
a. 1
b. 1/n
c. n
d. (1-n)/n
e. 1- 1/n
nvm i got it, its 1/n
if f(x)=n√x = n (x)^(1/2)
f'(x) = (1/2)n x^(-1/2)
= n/(2√x)
f'(1) = n/2 <------ none of the given choices
To find the derivative of f(x) = n√x or x^(1/n), we can use the power rule of differentiation.
The power rule states that if we have a function of the form f(x) = x^k, then the derivative of f(x) with respect to x is given by f'(x) = kx^(k-1).
In this case, we have f(x) = x^(1/n), where n is a constant. Applying the power rule, we get:
f'(x) = (1/n)x^((1/n)-1)
To find f'(1), we substitute x = 1 into the derived expression:
f'(1) = (1/n)(1^((1/n)-1))
Simplifying the expression:
f'(1) = (1/n)(1^(1/n) * 1^(-1))
Recall that any number raised to the power of 0 is equal to 1:
f'(1) = (1/n)(1 * 1^-1)
Using the property that any nonzero number raised to the power of -1 is its reciprocal:
f'(1) = (1/n)(1/1)
Simplifying further:
f'(1) = 1/n
Therefore, the correct answer is option b. 1/n.