Problem 67 - Calculus for Biology and Medicine

Question

Differentiate with respect to the independent variable. \(g(s)=\frac{s^{1 / 3}-1}{s^{2 / 3}-1}\)

Step-by-Step Solution

Answer

The derivative is \( g'(s) = \frac{\frac{1}{3}(s^{-2/3} - 1)}{(s^{2/3} - 1)^2} \).

Step 1: Identify the format of the function

The function given is in the format of a quotient: \( g(s) = \frac{u(s)}{v(s)} \) where \( u(s) = s^{1/3} - 1 \) and \( v(s) = s^{2/3} - 1 \). This suggests using the quotient rule for differentiation.

Step 2: Recall the quotient rule

The quotient rule is used to differentiate functions that are ratios. If \( g(s) = \frac{u(s)}{v(s)} \), then the derivative \( g'(s) \) is given by: \[ g'(s) = \frac{u'(s)v(s) - u(s)v'(s)}{(v(s))^2} \].

Step 3: Differentiate the numerator and the denominator

Find \( u'(s) \) and \( v'(s) \) separately:- \( u(s) = s^{1/3} - 1 \) - \( u'(s) = \frac{1}{3}s^{-2/3} \)- \( v(s) = s^{2/3} - 1 \) - \( v'(s) = \frac{2}{3}s^{-1/3} \).

Step 4: Apply the quotient rule

Substitute the derivatives found in Step 3 into the quotient rule:\[ g'(s) = \frac{\left(\frac{1}{3}s^{-2/3}\right)(s^{2/3} - 1) - (s^{1/3} - 1)\left(\frac{2}{3}s^{-1/3}\right)}{(s^{2/3} - 1)^2} \].

Step 5: Simplify the expression

Carefully distribute terms and simplify:- \( \left(\frac{1}{3}s^{-2/3}\right)(s^{2/3} - 1) = \frac{1}{3} - \frac{1}{3}s^{-2/3} \),- \( (s^{1/3} - 1)\left(\frac{2}{3}s^{-1/3}\right) = \frac{2}{3} - \frac{2}{3}s^{-2/3} \).Combine:\[ g'(s) = \frac{\left(\frac{1}{3} - \frac{1}{3}s^{-2/3}\right) - \left(\frac{2}{3} - \frac{2}{3}s^{-2/3}\right)}{(s^{2/3} - 1)^2} \].Further simplification:\[ g'(s) = \frac{-\frac{1}{3} + \frac{1}{3}s^{-2/3}}{(s^{2/3} - 1)^2} \].Thus,\[ g'(s) = \frac{\frac{1}{3}(s^{-2/3} - 1)}{(s^{2/3} - 1)^2} \].

Key Concepts

Quotient Rule

The quotient rule is a handy tool in calculus when dealing with the derivative of functions expressed as the ratio of two differentiable functions. For a function written as a fraction

• numerator: \( u(s) \)
• denominator: \( v(s) \)

Differentiating using the quotient rule involves these steps:

• Find the derivative of the numerator \( u'(s) \)
• Find the derivative of the denominator \( v'(s) \)
• Plug these derivatives into the quotient formula: \[ g'(s) = \frac{u'(s)v(s) - u(s)v'(s)}{(v(s))^2} \]

This rule helps avoid complex manipulation and provides a straightforward way to differentiate quotients without unnecessary steps. It's like a shortcut to more efficiently finding derivatives of fractions.

Derivative Calculation

In calculus, calculating derivatives accurately is crucial. For our example function \( g(s) = \frac{s^{1/3} - 1}{s^{2/3} - 1} \), it's essential to calculate both the numerator's and the denominator's derivatives separately.
The key is to:

• Calculate \( u'(s) \) for the numerator, where \( u(s) = s^{1/3} - 1 \). Here, using the power rule, we find \( u'(s) = \frac{1}{3}s^{-2/3} \).

Also, for the denominator:

• Determine \( v'(s) \) where \( v(s) = s^{2/3} - 1 \). The derivative there is \( v'(s) = \frac{2}{3}s^{-1/3} \).

These derivatives plug directly into the quotient rule, making our differentiation process seamless and systematic.

Algebraic Simplification

After applying the quotient rule to find the derivative, simplifying the expression is crucial for a tidy, understandable result. During simplification:

• Distribute and collect like terms carefully. For example, distributing terms in our problem leads to: \[ \left(\frac{1}{3}s^{-2/3}\right)(s^{2/3} - 1) = \frac{1}{3} - \frac{1}{3}s^{-2/3} \]

Similarly, with the other part, ensure terms are correctly combined:

• \( (s^{1/3} - 1)\left(\frac{2}{3}s^{-1/3}\right) = \frac{2}{3} - \frac{2}{3}s^{-2/3} \)

After combining terms, check the expression for common factors to cancel or combine, providing a cleaner final form. The ultimate goal is to clearly present the derivative, showcasing the elegant behavior of the function's change rate.