find the derivative of f, when f(x)=1/sqrt(x)
ok so i tried simplifying it by multiplying top and bottom by conjugate radical, the top simplified to h but the bottom was a mess.
To find the derivative of the function f(x) = 1/√(x), we can use the quotient rule.
But first, let's simplify the expression.
To eliminate the square root, multiply the numerator and denominator by √(x) to get:
f(x) = 1/√(x) * √(x)/√(x) = √(x)/x
Now, let's find the derivative using the quotient rule:
The quotient rule states that if you have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x)*h(x) - g(x)*h'(x))/(h(x))^2
In our case, g(x) = √(x), and h(x) = x. So, we have:
f'(x) = (√(x)'*x - √(x)*x')/(x)^2
To find the derivative of √(x), we can use the power rule:
(√(x))' = (1/2)x^(-1/2)
Taking the derivative of x gives us x' = 1.
Now, let's substitute these values into the quotient rule formula:
f'(x) = ((1/2)x^(-1/2)*x - √(x)*1)/(x)^2
Simplifying:
f'(x) = (1/2)x^(1/2 - 2) - √(x)/(x)^2
f'(x) = (1/(2x^(3/2))) - √(x)/(x)^2
Therefore, the derivative of f(x) = 1/√(x) with respect to x is:
f'(x) = (1/(2x^(3/2))) - √(x)/(x)^2