find f'(x) for f(x) = x/(sqrtx^2+1) please show all the steps. I'm realy confused on this one. I get as far as setting it up using the quotient rule then I go blank. thanks
I will assume you meanst
f(x) = x/√(x^2+1) or x/(x^2 + 1)^(1/2)
did you have
f'(x) = [(x^2 + 1)^(1/2) - x(1/2)(x^2 + 1)^(-1/2)(2x)]/(x^2 + 1) ?
take out a common factor of (x^2+1)^(-1/2) from the numerator
= (x^2 + 1)^(-1/2) [ x^2 + 1 - x^2] / (x^2 + 1)
= 1/(x^2 + 1)^(3/2) or (x^2 + 1)^(-3/2)
THANK YOU!!!!!!
To find the derivative of the function f(x) = x/(√(x^2 + 1)), we can use the quotient rule. The quotient rule states that if we 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
Now, let's find the derivative step by step:
Step 1: Identify the numerator and the denominator functions.
In this case, g(x) = x and h(x) = √(x^2 + 1).
Step 2: Find the derivatives of g(x) and h(x).
g'(x) = 1 (since the derivative of x with respect to x is 1)
h'(x) = [(1/2) * (x^2 + 1)^(-1/2)] * 2x = x / √(x^2 + 1)
Step 3: Apply the quotient rule formula.
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Substituting the derivatives we found:
f'(x) = (1 * √(x^2 + 1) - x * (x / √(x^2 + 1))) / (√(x^2 + 1))^2
Simplifying the expression:
f'(x) = (√(x^2 + 1) - x^2 / √(x^2 + 1)) / (x^2 + 1)
To further simplify, multiply the numerator by √(x^2 + 1) / √(x^2 + 1):
f'(x) = (√(x^2 + 1) - (x^2 / √(x^2 + 1))) * √(x^2 + 1) / (x^2 + 1)
Expanding the numerator:
f'(x) = (x^2 + 1 - x^2) / [(x^2 + 1) * √(x^2 + 1)]
Simplifying:
f'(x) = 1 / √(x^2 + 1)
So, the derivative of f(x) = x/(√(x^2 + 1)) is f'(x) = 1 / √(x^2 + 1).
Remember to check if there are any restrictions or conditions for the domain of the function or any points of discontinuity.