Second derivatives Find d²y/dx²for the following functions.
y = e^-2x²

Verified step by step guidance

1. Step 1: Identify the function y = e^{-2x^2}. This is an exponential function where the exponent is a function of x, specifically -2x^2.
2. Step 2: Find the first derivative, dy/dx. Use the chain rule, which states that if y = e^u, then dy/dx = e^u * du/dx. Here, u = -2x^2, so du/dx = -4x.
3. Step 3: Apply the chain rule to find dy/dx = e^{-2x^2} * (-4x) = -4x * e^{-2x^2}.
4. Step 4: Find the second derivative, d²y/dx². Differentiate dy/dx = -4x * e^{-2x^2} using the product rule, which states that if u and v are functions of x, then d(uv)/dx = u'v + uv'.
5. Step 5: Apply the product rule: Let u = -4x and v = e^{-2x^2}. Then, u' = -4 and v' = e^{-2x^2} * (-4x) (using the chain rule again). Substitute these into the product rule formula to find d²y/dx².

Key Concepts

Second Derivative

The second derivative of a function 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. Mathematically, it is denoted as d²y/dx² and is calculated by differentiating the first derivative.

Exponential Functions

Exponential functions are mathematical expressions in the form y = a * e^(bx), where e is Euler's number (approximately 2.71828). In the context of the given function y = e^(-2x²), the exponent is a quadratic expression, which affects the growth and decay behavior of the function. Understanding the properties of exponential functions is crucial for differentiating them correctly.

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. This rule is essential for finding derivatives of functions like y = e^(-2x²), where the exponent is a function of x.