Problem 33 - Calculus for Biology and Medicine

Question

Differentiate $$ g(T)=a\left(T_{0}-T\right)^{3}-b $$ with respect to \(T\). Assume that \(a, b\), and \(T_{0}\) are positive constants.

Step-by-Step Solution

Answer

The derivative of \(g(T)=a(T_{0}-T)^{3}-b\) with respect to \(T\) is \(-3a(T_0 - T)^2\).

Step 1: Identify the Differentiation Rule Needed

To differentiate the function \(g(T)=a(T_{0}-T)^{3}-b\), we need to apply the chain rule. The chain rule is needed because \((T_{0}-T)^{3}\) is a composition of the outer function \(x^3\) and the inner function \((T_{0} - T)\).

Step 2: Differentiate the Outer Function Using Chain Rule

Identify the outer function as \( (T_{0} - T)^3 \). Applying the differentiation rule for \(x^n\), we first differeniate with respect to \((T_{0} - T)\). The derivative of \((T_{0} - T)^3\) with respect to \((T_{0} - T)\) is \(3(T_0 - T)^2\).

Step 3: Differentiate the Inner Function

The inner function is \((T_0 - T)\). Differentiate this with respect to \(T\) to get its derivative. The derivative of \(T_0 - T\) with respect to \(T\) is \(-1\).

Step 4: Apply the Chain Rule

Apply the chain rule by multiplying the derivative of the outer function \(3(T_0 - T)^2\) by the derivative of the inner function \(-1\), giving \(-3(T_0 - T)^2\).

Step 5: Apply Constant Multiplication

Since the original function is multiplied by a constant \(a\), apply the constant multiplication rule. Multiply the result from Step 4 by \(a\), resulting in \( -3a(T_0 - T)^2 \).

Step 6: Differentiate the Constant Term

Differentiate the constant term \(-b\). The derivative of a constant is zero, so this does not contribute to the derivative of the function.

Step 7: Combine Results

Combine the results from Steps 5 and 6 to write out the final derivative of the function. The derivative of \(g(T)=a(T_{0}-T)^{3}-b\) with respect to \(T\) is \(-3a(T_0 - T)^2\).

Key Concepts

Understanding the Chain Rule

In calculus, the chain rule is a powerful tool for differentiating composite functions. A composite function is simply a function within another function, like \(f(g(x))\). To understand it better, consider \(g(T) = a(T_{0} - T)^{3} - b\). Here, the 'inner function' is \(T_{0} - T\) and the 'outer function' is \(x^3\) where you replace \(x\) with the inner function.
This nested structure is where the chain rule comes into play.

• Step 1: Differentiate the outer function (the cube operation in \(x^3\)). So, the derivative of \(u^3\) with respect to \(u\) is \(3u^2\).
• Step 2: Differentiate the inner function \(T_{0} - T\). The derivative with respect to \(T\) is \(-1\).
• Step 3: Multiply these derivatives together as per the chain rule: the result is \(-3(T_{0} - T)^{2}\).

This process allows you to differentiate complex, nested functions easily by breaking them down into simpler steps.

Applying Constant Multiplication

Constant multiplication is straightforward but essential in calculus. It states that when a function is multiplied by a constant, its derivative will also be multiplied by that same constant. This concept simplifies differentiation when constants are present in functions.
Going back to the function \(g(T) = a(T_{0} - T)^{3} - b\), the term \(a(T_{0} - T)^{3}\) involves constant multiplication because \(a\) is a constant multiplier.Here's how you apply it:

• Find the derivative of \( (T_{0} - T)^3 \) using the chain rule, which we found to be \(-3(T_{0} - T)^2\).
• Multiply this result by the constant \(a\), giving \(-3a(T_{0} - T)^2\).
By understanding constant multiplication, you can effortlessly handle derivatives that include constant factors, simplifying part of your calculations.

Effective Calculus Problem Solving

Problem-solving in calculus requires a systematic approach to break down complex tasks into manageable steps. It is important to recognize the type of differentiation involved, whether chain rule, product rule, or any other rule. Let's revisit the problem from above:
The initial function you had was \(g(T) = a(T_{0} - T)^{3} - b\). Notice how the multi-step solution was achieved by:

• Identifying the need for the chain rule due to the composite structure of the function.
• Handling constant terms using the constant multiplication principle.
• Understanding that constants like \(-b\) yield zero when differentiated, simplifying the task.

Effective calculus problem-solving involves:

• Identifying the problem type and relevant rules.
• Breaking the problem into simpler parts (like handling inner and outer functions separately).
• Applying rules systematically and carefully combining the results for accuracy.

These steps enhance fluency in calculus and equip you with the skills to tackle more challenging problems in the future.