Problem 68 - Calculus for Biology and Medicine

Question

Differentiate with respect to the independent variable. \(g(s)=\frac{s^{1 / 7}-s^{2 / 7}}{s^{3 / 7}+s^{4 / 7}}\)

Step-by-Step Solution

Answer

Differentiate using the quotient rule: \( g'(s) = \frac{(u'v - uv')}{v^2} \).

Step 1: Identify the Function

The function we need to differentiate is given as: \[ g(s) = \frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}} \]This is a rational function where both the numerator and denominator are polynomials in terms of \( s \).

Step 2: Apply the Quotient Rule

For a function \( f(x) = \frac{u(x)}{v(x)} \), the derivative using the Quotient Rule is: \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \] In our case, \( u(s) = s^{1/7} - s^{2/7} \) and \( v(s) = s^{3/7} + s^{4/7} \).

Step 3: Differentiate the Numerator

Differentiate \( u(s) = s^{1/7} - s^{2/7} \):\[ u'(s) = \frac{1}{7}s^{-6/7} - \frac{2}{7}s^{-5/7} \] This uses the power rule for differentiation.

Step 4: Differentiate the Denominator

Differentiate \( v(s) = s^{3/7} + s^{4/7} \):\[ v'(s) = \frac{3}{7}s^{-4/7} + \frac{4}{7}s^{-3/7} \] Again, apply the power rule to each term.

Step 5: Substitute into the Quotient Rule Formula

Substitute \( u(s) \), \( u'(s) \), \( v(s) \), and \( v'(s) \) into the quotient rule:\[ g'(s) = \frac{(\frac{1}{7}s^{-6/7} - \frac{2}{7}s^{-5/7})(s^{3/7} + s^{4/7}) - (s^{1/7} - s^{2/7})(\frac{3}{7}s^{-4/7} + \frac{4}{7}s^{-3/7})}{(s^{3/7} + s^{4/7})^2} \]

Step 6: Simplify the Expression

Carefully expand the terms in the numerator and combine like terms. Simplification involves lengthy algebraic steps, focusing on distributing and combining like powers of \( s \).

Key Concepts

Quotient Rule

The Quotient Rule is a vital tool in calculus when differentiating functions expressed as one function divided by another, specifically rational functions. You can recall the formula for the Quotient Rule by remembering that it involves differentiation of both the numerator and the denominator. If you have a function of the form \( f(x) = \frac{u(x)}{v(x)} \), the derivative is given by the formula:

• \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \)

This approach requires you to:

• Find \( u(x) \) and \( v(x) \), which are the numerator and the denominator, respectively.
• Differentiate both \( u(x) \) and \( v(x) \) to get \( u'(x) \) and \( v'(x) \).
• Substitute these into the Quotient Rule formula.

By using this method, you can effectively handle complex rational expressions like \( g(s) = \frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}} \). Always note the operations involve both multiplication between derivatives and functions, and subtraction of their products. Consistency in applying this step-by-step pattern is key to success.

Rational Functions

Rational functions are expressions that consist of one polynomial function divided by another. Hence, they take the general form \( R(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. Understanding these functions requires recognizing the behavior of both the numerator and the denominator independently and together.
Handling rational functions often leads to using more advanced calculus tools such as the Quotient Rule. In the exercise, the rational function presented is \( g(s) = \frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}} \). In this function:

• The numerator \( s^{1/7} - s^{2/7} \) highlights subtraction of two terms expressed in terms of \( s \).
• The denominator \( s^{3/7} + s^{4/7} \) showcases an addition of terms, each a power of \( s \).

When differentiating rational functions, examining each component of \( s \)'s powers is vital to properly applying appropriate differentiation rules. The rational structure necessitates care, ensuring that terms relating to division are handled correctly, avoiding common pitfalls such as division by zero when evaluating the function.

Power Rule for Differentiation

The Power Rule is perhaps the simplest yet most crucial rule for differentiating powers of variables. It fundamentally states that for any term in the form of \( x^n \), the derivative is \( nx^{n-1} \). This is used to differentiate terms in the numerator and the denominator of the rational function given in the problem:

• For \( s^{1/7} \), using the Power Rule gives you \( \frac{1}{7}s^{-6/7} \).
• Similarly, \( s^{2/7} \) becomes \( \frac{2}{7}s^{-5/7} \).
• For the denominator, \( s^{3/7} \) turns into \( \frac{3}{7}s^{-4/7} \), and \( s^{4/7} \) changes to \( \frac{4}{7}s^{-3/7} \).

To apply the Power Rule effectively:

• Determine the exponent of each term.
• Multiply the base by this exponent minus one.
• Simplify the result to obtain the derivative of each power involved.

In rational functions, power differentiation significantly aids in determining the derivatives of complex expressions, as evidenced in our problem. Mastering the Power Rule promises smooth sailing in calculus, especially in tackling rational expressions efficiently.