Find the first and second derivatives of the function.
h(x) = sqrt x^2 + 1
(x^2+1)^.5
y'= .5 (x^2+1)^-.5 * (2x)
y"= I will leave that to you.
To find the first and second derivatives of the function h(x) = sqrt(x^2 + 1), we'll use the power rule and the chain rule.
1. First Derivative:
The power rule states that the derivative of x^n is n * x^(n-1). Let's start by simplifying the function:
h(x) = sqrt(x^2 + 1) = (x^2 + 1)^(1/2)
Now, applying the chain rule, we have:
h'(x) = (1/2) * (x^2 + 1)^(-1/2) * (2x)
= x / sqrt(x^2 + 1)
So, the first derivative of h(x) is h'(x) = x / sqrt(x^2 + 1).
2. Second Derivative:
To find the second derivative, we'll differentiate h'(x) using the quotient rule.
The quotient rule states that if we have a function f(x) = g(x) / h(x), then f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.
Applying the quotient rule to h'(x) = x / sqrt(x^2 + 1), we have:
h''(x) = [(1) * sqrt(x^2 + 1) - (x) * (1/2) * (x^2 + 1)^(-1/2) * (2x)] / (x^2 + 1)
= [sqrt(x^2 + 1) - (x^2)/(sqrt(x^2 + 1))] / (x^2 + 1)
= (1 - x^2) / (x^2 + 1)^(3/2)
Therefore, the second derivative of h(x) is h''(x) = (1 - x^2) / (x^2 + 1)^(3/2).