7–28. Derivatives Evaluate the following derivatives.


d/dx (e^{-10x²})

Verified step by step guidance

1. Step 1: Recognize that the function involves the exponential function e raised to a power, specifically e^{-10x²}. The derivative of e^u with respect to x is e^u * du/dx, where u is a function of x.
2. Step 2: Identify the inner function u = -10x². To apply the chain rule, we need to compute the derivative of u with respect to x.
3. Step 3: Differentiate u = -10x² with respect to x. Using the power rule, the derivative of x² is 2x, and multiplying by the constant -10 gives du/dx = -20x.
4. Step 4: Apply the chain rule. The derivative of e^{-10x²} with respect to x is e^{-10x²} * (-20x), combining the derivative of the outer function e^u and the inner function u = -10x².
5. Step 5: Simplify the expression. The derivative is -20x * e^{-10x²}. This is the simplified form of the derivative.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. Understanding how to compute derivatives is essential for analyzing the behavior of functions, including their maxima, minima, and points of inflection.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of the natural logarithm, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay, depending on the sign of 'b'. In the context of derivatives, the derivative of an exponential function is proportional to the function itself, which simplifies the differentiation process.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions that involve exponentials, as seen in the differentiation of e^{-10x²}.