Second derivatives Find d²y/dx²for the following functions.
y = √x²+2

Verified step by step guidance

1. First, identify the function y = √(x² + 2). To find the second derivative, we need to first find the first derivative dy/dx.
2. Use the chain rule to differentiate y = (x² + 2)^(1/2). The chain rule states that if you have a composite function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
3. Differentiate the outer function f(u) = u^(1/2) with respect to u, which gives f'(u) = (1/2)u^(-1/2). Then, differentiate the inner function g(x) = x² + 2 with respect to x, which gives g'(x) = 2x.
4. Apply the chain rule: dy/dx = (1/2)(x² + 2)^(-1/2) * 2x. Simplify this expression to find the first derivative.
5. To find the second derivative d²y/dx², differentiate the first derivative with respect to x. Use the product rule and chain rule as necessary, and simplify the resulting expression.

Key Concepts

Second Derivative

The second derivative of a function, denoted as d²y/dx², measures the rate of change of the first derivative. It provides information about the concavity of the function and can indicate points of inflection where the function changes from concave up to concave down or vice versa.

Chain Rule

The chain rule is a fundamental differentiation technique used when differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x.

Implicit Differentiation

Implicit differentiation is a method used to differentiate equations where y is not explicitly solved for x. This technique allows us to find derivatives of y with respect to x by treating y as a function of x, even when y is defined implicitly through an equation involving both x and y.