91Ó°ÊÓ

The chain rule is a fundamental principle in calculus for finding the derivative of a composite function. Imagine a function composed of two or more nested functions. The chain rule allows us to differentiate each part individually and then combine them. This rule can be summarized by the formula: if we have two functions, say, \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \).
This means we first take the derivative of the outer function \( f \) with respect to its input \( g(x) \), and then multiply it by the derivative of the inner function \( g \) with respect to \( x \).

• Outer function: Evaluate the derivative of the outer function but keep the inner function intact.
• Inner function: Now, evaluate the derivative of the inner function.

In our example, the function \( w = 100 e^{-x^2} \) requires the chain rule. Here, the exponential function is the outer function, and the quadratic \( -x^2 \) is the inner function. The chain rule helps us combine these derivatives efficiently to get the final expression.

Exponential functions are functions of the form \( e^{x} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions have unique properties and play a crucial role in calculus, especially in growth and decay models.

• The derivative of \( e^x \) is simply \( e^x \), which means the rate of change of \( e^x \) is itself.
• Exponential functions like \( e^{-x^2} \) can represent more complex expressions and have a profound impact on solving calculus problems, especially when combined with other functions using rules like the chain rule.

In the problem we tackled, \( w = 100 e^{-x^2}\), we treated \( 100 e^u \) as our exponential function. By differentiating with respect to its input \( u \), we got \( 100 e^u \), and the constant 100 was simply carried through the differentiation process.

The power rule is a shortcut for finding the derivative of functions of the form \( x^n \), where \( n \) is a real number. This rule states that the derivative of \( x^n \) is \( nx^{n-1} \). It's particularly handy for polynomials.
Here's how it is typically applied:

• Take the exponent \( n \) and multiply it by the coefficient of \( x \).
• Subtract one from the original exponent to get the new exponent of \( x \).

In our exercise, we used the power rule to find the derivative of the inner function \( u(x) = -x^2 \). Here, \( n = 2 \), making the derivative \( -2x \). This step was essential prior to applying the chain rule, as it allowed us to recognize the contribution of the inner function to the composite function's overall derivative.