91Ó°ÊÓ

In differentiation, the chain rule is a fundamental technique used to find the derivative of a composite function. Think of a composite function as a function within another function. The chain rule helps us differentiate these composite constructions by focusing on the relationships between the inner and outer functions.

Let's break down the process further with an example:
\ Given a function \(f(x) = g(h(x))\), the chain rule states that the derivative of \(f(x)\) is \(\frac{d}{dx} [g(h(x))] = g'(h(x)) \times h'(x)\).

This simply means we first take the derivative of the outer function \(g(x)\) and keep the inner function \(h(x)\) intact. Then, we multiply this by the derivative of the inner function \(h(x)\).

Applying this to step 3 of our problem:
The given function is \(f(x) = e^{-x/2}\). Here, \(g(u) = e^u\) and \(h(x) = -x/2\).

Therefore, \(g'(u) = e^u\) and \(h'(x) = -1/2\).

Using the chain rule:
\[ f'(x) = g'(h(x)) \times h'(x) = e^{-x/2} \times (-1/2) = -\frac{1}{2}e^{-x/2} \]

This step-by-step breakdown provides a clear appreciation of how the chain rule simplifies the differentiation of composite functions.

Exponential functions are a critical topic in calculus and mathematics in general. They are functions of the form \(f(x) = a^{x}\) where 'a' is a constant.

A very common exponential function is one with the base 'e', known as the natural exponential function, written as \(e^x\). The number 'e' is an irrational constant approximately equal to 2.71828.

Exponential functions have unique properties, such as their rate of growth being proportional to their value. This makes them particularly useful in modeling real-world phenomena such as population growth, radioactive decay, and interest rates.

In our problem, we have the function \(e^{-x/2}\), an exponential function where the exponent itself (\