91Ó°ÊÓ

The chain rule is a fundamental tool in calculus used for differentiating composite functions. A composite function is a function that arises when one function is inside another function. When you have a composite function like this, the chain rule helps us find the derivative by breaking down the process into manageable steps.
To use the chain rule, imagine we have a function \( F(x) = g(h(x)) \). The derivative of \( F \) with respect to \( x \) is found by taking the derivative of \( g \) with respect to \( h \), and then multiplying it by the derivative of \( h \) with respect to \( x \). In mathematical terms, this is expressed as:

• \( \frac{d}{dx} F(x) = \frac{dg}{dh} \cdot \frac{dh}{dx} \)

In our problem, the function \( F(z) = \left(\frac{z-1}{z+1}\right)^{1/2} \) required the chain rule because it involved an outer function \( u^{1/2} \) where \( u = \frac{z-1}{z+1} \). We needed to find the derivative of the whole while focusing on \( u \) first then \( z \).The chain rule is powerful because it simplifies the process of differentiating complex functions into simpler tasks. Just remember to tackle the function from the outside in, using the derivative of each layer as it peels away.

The power rule is one of the simplest and most widely used rules in calculus for differentiation. It makes finding the derivative of a polynomial or a function with powers straightforward. The power rule states that if you have a function of the form \( x^n \), where \( n \) is any real number, then the derivative is:

• \( \frac{d}{dx} x^n = n \cdot x^{n-1} \)

In our exercise, the function \( F(z) = \left(\frac{z-1}{z+1}\right)^{1/2} \) was transformed into a form more suitable for differentiation by using the power rule. The term \( \left(\frac{z-1}{z+1}\right)^{1/2} \) can be treated as \( u^{1/2} \) after letting \( u = \frac{z-1}{z+1} \).
Applying the power rule to differentiate this, we get:

• \( \frac{d}{du} u^{1/2} = \frac{1}{2} u^{-1/2} \)

This result was then multiplied by the derivative of \( u \) with respect to \( z \), according to the chain rule. In problems involving powers, always look for an opportunity to use the power rule as it will simplify finding derivatives.
It's a straightforward technique that saves a lot of time, especially when dealing with polynomial terms or expressions with fractional exponents.

The quotient rule is essential for differentiating functions that are expressed as one function divided by another, essentially a fraction. When tackling a function in the form of \( g(x)/h(x) \), the quotient rule provides a formula for finding its derivative.The quotient rule is given by:

• \( \frac{d}{dx} \frac{g(x)}{h(x)} = \frac{g'(x) h(x) - g(x) h'(x)}{(h(x))^2} \)

In our specific exercise, the inner function \( u = \frac{z - 1}{z + 1} \) required the quotient rule for its differentiation. By applying the quotient rule, we find:

• \( \frac{du}{dz} = \frac{(z+1)(1) - (z-1)(1)}{(z+1)^2} = \frac{2}{(z+1)^2} \)

Note that the numerator results from multiplying the derivative of the top expression by the bottom expression and subtracting the product of the top expression by the derivative of the bottom.
The quotient rule is perfect for differentiating fractional functions because it systematically handles both the numerator and denominator, ensuring all components are considered. It's crucial to keep track of each part of the fraction and remember to square the bottom function in the denominator, ensuring accuracy in your calculations.