If y=sqrt (x^2+16), then d^2y/dx^2=
A. 1 / (4(x^2+16)^3/2)
B. 4(3x^2+16)
C. x / ((x^2+16)^1/2)
D. (2x^2+16) / ((x^2+16)^3/2)
E. 16 / ((x^2+16)^3/2)
To find the second derivative of y with respect to x, we need to differentiate the expression dy/dx, which is the first derivative of y with respect to x. Let's start by finding the first derivative.
Given: y = √(x^2 + 16)
To find dy/dx, we can use the chain rule. The chain rule states that if we have a function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x), where f'(x) represents the derivative of f(x), and g'(x) represents the derivative of g(x).
In this case, f(g) = √g and g(x) = x^2 + 16. Applying the chain rule, we get:
dy/dx = (1/2)(x^2 + 16)^(-1/2)(2x)
Simplifying, we have:
dy/dx = x / (x^2 + 16)^1/2
Now, to find the second derivative, d^2y/dx^2, we differentiate dy/dx with respect to x.
Using the quotient rule, which states that if we have a function f(x) / g(x), then (f/g)' = (f'g - fg') / g^2, we can find the second derivative.
In this case, f(x) = x and g(x) = (x^2 + 16)^1/2. Applying the quotient rule, we get:
d^2y/dx^2 = [(1)(x^2 + 16)^1/2 - (x)(x / (x^2 + 16)^1/2)] / (x^2 + 16)
Simplifying, we have:
d^2y/dx^2 = (x^2 + 16 - x^2) / (x^2 + 16)^(3/2)
d^2y/dx^2 = 16 / (x^2 + 16)^(3/2)
Therefore, the correct answer is E. 16 / ((x^2+16)^3/2).