Problem 14 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Let \(P(t)=u(t) / v(t)\). Then $$ \ln P(t)=\ln \left(\frac{u(t)}{v(t)}\right)=\ln u(t)-\ln v(t) $$ Compute the derivative of the two sides of Equation 6.12 using the logarithm chain rule and show that $$ P^{\prime}(t)=\frac{u(t) v^{\prime}(t)-u^{\prime}(t) v(t)}{v^{2}(t)} $$

Step-by-Step Solution

Answer

Differentiate both sides; equate derivatives and solve for \( P'(t) \).

Step 1: Differentiate the Left Side

Firstly, differentiate the left side of the equation \( \ln P(t) \) with respect to \( t \). This gives \( \frac{d}{dt}[\ln P(t)] = \frac{P'(t)}{P(t)} \) as per the chain rule for derivatives of natural logarithms.

Step 2: Differentiate the Right Side

Differentiate the right side \( \ln u(t) - \ln v(t) \) with respect to \( t \). Using the chain rule, you get \( \frac{d}{dt}[\ln u(t)] - \frac{d}{dt}[\ln v(t)] = \frac{u'(t)}{u(t)} - \frac{v'(t)}{v(t)} \).

Step 3: Relate the Derivatives

Since the derivative of the left side must equal the derivative of the right side, we equate: \( \frac{P'(t)}{P(t)} = \frac{u'(t)}{u(t)} - \frac{v'(t)}{v(t)} \).

Step 4: Substitute and Simplify

Substitute \( P(t) = \frac{u(t)}{v(t)} \) into the equation from Step 3, yielding \( \frac{P'(t)}{\frac{u(t)}{v(t)}} = \frac{u'(t)}{u(t)} - \frac{v'(t)}{v(t)} \). This simplifies to \( P'(t) = \left( \frac{v(t)u'(t) - u(t)v'(t)}{v^2(t)} \right) \), using algebraic manipulation and simplification.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation simplifies the process of differentiating complex functions, especially those involving products or quotients. This technique involves taking the natural logarithm of both sides of an equation and then differentiating. By doing so, it transforms multiplicative relationships into additive ones, making the differentiation process more manageable.

Here's a simple procedure for using logarithmic differentiation:

• Apply the natural logarithm to both sides of the given equation.
• Simplify the logarithmic expression using logarithmic properties, such as \( \ln(a/b) = \ln a - \ln b \).
• Differentiate both sides of the equation.
• Solve for the desired derivative.

This technique is particularly useful for functions in the form of fractions, like \( P(t) = \frac{u(t)}{v(t)} \). It utilizes the power of logarithms to make the differentiation simple and straightforward.

Quotient Rule

The quotient rule is an essential tool in calculus for finding the derivative of a quotient of two differentiable functions. Simply put, if you have a function \( P(t) = \frac{u(t)}{v(t)} \), where both \( u(t) \) and \( v(t) \) are differentiable, the quotient rule allows us to compute its derivative efficiently.

The formula for the quotient rule is:\[P'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}\]This result arises from combining the derivatives of both the numerator and the denominator in a specific way:

• First, take the derivative of the top function \( u(t) \) and multiply it by the bottom function \( v(t) \).
• Next, subtract the product of the derivative of the bottom function \( v(t) \) with the top function \( u(t) \).
• Finally, divide the entire expression by the square of the bottom function \( (v(t))^2 \).

This procedure is particularly straightforward when you have already used logarithmic differentiation to simplify the original problem.

Calculus Problem Solving

Problem solving in calculus often involves breaking down the problem, using appropriate differentiation rules, and simplifying. Calculus provides us with various tools and rules, like the quotient rule and logarithmic differentiation, to tackle complex functions systematically.

When solving a calculus problem:

• Identify the type of function you are dealing with, such as a product or a quotient.
• Select the most suitable differentiation technique. For example, use logarithmic differentiation for functions involving products or quotients.
• Apply the differentiation rule carefully to ensure accuracy in deriving expressions.
• Simplify the obtained derivative to make it more interpretable.

By following these steps, one can systematically and accurately solve calculus problems, ensuring all steps contribute to the final solution.

Chain Rule in Calculus

The chain rule is a fundamental principle in calculus used to differentiate compositions of functions. It allows derivative calculations when dealing with nested functions, by relating their derivatives in a systematic manner.

To apply the chain rule, consider if you have a composite function \( f(g(t)) \). It's described by:\[\frac{d}{dt}[f(g(t))] = f'(g(t)) \, g'(t)\]Here's how you can think of it:

• First, differentiate the outer function \( f \), leaving the inner function \( g(t) \) inside.
• Then, multiply by the derivative of the inner function \( g(t) \).

This approach is particularly useful when combined with logarithmic differentiation, as seen in differentiating \( \ln P(t) \), where the chain rule helps manage the nested logarithmic functions.