What's the derivative of this function? Question #13

1.)
f(x) = 1 / sqrt(2x-1)
sqrt means square root



Please help me find the derivatives and please show work too! 10 points to most accurate answer!

You can use the quotient rule, but I prefer the product rule easier to use and easier to remember. You do this by bringing the denominator up next to the numerator, and turning it into a negative exponent. For example, 2 / x^2 can be written as 2x^-2 to use the product rule on. Also, a number like √x means x^(1/2).

f(x) = 1 / √(2x-1)
= 1 / (2x-1)^(1/2)
= 1 * (2x-1)^(-1/2)
= (2x-1)^(-1/2)

Well in this case, we didn't even need to use the product rule because the numerator happened to be simply 1 rather than something more complex like x^4.

f'(x) = (-1/2)(2x-1)^(-3/2) * 2
= (-2/2)(2x-1)^(-3/2)
= (-1)(2x-1)^(-3/2)
= -1 / (2x-1)^(3/2)

let u = 2x-1

f(u) = 1/√(u) or u^-(1/2)

now find f'(u) = -(1/2)u^-(3/2) * du/dx

but du/dx = d(2x-1)/dx which is 2

the (1/2) and 2 cancel out

leaving -u^-(3/2)

but u is 2x-1

so we have -(2x-1)^-(3/2)

or put it another way

- 1/√(2x-1)^3

f(x) = (2x - 1)^(-1/2)

f '(x) = -1/(2x - 1)^(3/2)

f(x) = (2x - 1)^(-1/2)
f'(x) = -(2x - 1)^(-3/2)
f'(x) = -1 / √(2x - 1)^3