Problem 36 - Calculus for Biology and Medicine

Question

Suppose that \(f(2)=-4, g(2)=3, f^{\prime}(2)=1\), and \(g^{\prime}(2)=-2\). Find $$ \left(f^{2}+g^{2}\right)^{\prime}(2) $$

Step-by-Step Solution

Answer

The derivative \((f^2 + g^2)'(2) = -20\).

Step 1: Understand the Problem

We need to find the derivative of the function \( (f^2 + g^2)(x) \) at \( x = 2 \). This is represented as \( (f^2(x) + g^2(x))' \) evaluated at \( x = 2 \).

Step 2: Apply the Sum Rule of Differentiation

The derivative of a sum is the sum of the derivatives. Therefore, we have: \( (f^2(x) + g^2(x))' = (f^2(x))' + (g^2(x))' \).

Step 3: Use the Chain Rule for Derivatives

For each part, use the chain rule. We have: For \( f^2(x) \), use: \( (f^2(x))' = 2f(x)f'(x) \).For \( g^2(x) \), use: \( (g^2(x))' = 2g(x)g'(x) \).

Step 4: Substitute Known Values

Insert the given values into the derivative formulas:- For \( (f^2(x))' \) at \( x = 2 \): \( 2f(2)f'(2) = 2(-4)(1) = -8 \).- For \( (g^2(x))' \) at \( x = 2 \): \( 2g(2)g'(2) = 2(3)(-2) = -12 \).

Step 5: Calculate the Sum of Derivatives

Add the results from the derivatives:\( (f^2 + g^2)'(2) = -8 + (-12) = -20 \).

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate at which a function changes at any given point. Think of a derivative as a measure of how a small change in one variable can affect another. In simpler terms, if you imagine a curve representing a function, the derivative tells you the slope of that curve at any specific point. To find a derivative, you typically start with a function, say \( f(x) \). You try to determine what happens to \( f(x) \) as \( x \) changes ever so slightly, mathematically expressed as \( \Delta x \to 0 \). The derivative of \( f(x) \), often noted as \( f'(x) \) or \( \frac{df}{dx} \), provides a precise rate of change. In the context of our exercise, we use derivatives to find how the function \( f^2(x) + g^2(x) \) changes with respect to \( x \) at a specific point (here, \( x = 2 \)). We are interested in the derivative because it allows us to understand how the entire expression \( f^2 + g^2 \) behaves at this point.

Chain Rule

The chain rule is an essential tool in calculus for differentiating composite functions. A composite function is formed when one function is applied inside another. For instance, if you have \( f(x) = h(g(x)) \), you are dealing with a composite function. The chain rule allows us to differentiate such functions in a manageable way. It states that if you have a function \( f(x) = h(g(x)) \), then the derivative \( f'(x) \) is given by \( h'(g(x)) \cdot g'(x) \). This means that you take the derivative of the outer function (\( h \)) evaluated at the inner function (\( g(x) \)), and multiply it by the derivative of the inner function (\( g(x) \)).In our exercise, we used the chain rule to differentiate \( f^2(x) \) and \( g^2(x) \). We recognized both as composite functions: for \( f^2(x) \), think of the function as \( h(u) = u^2 \) with \( u = f(x) \). Hence, its derivative becomes \( 2f(x)f'(x) \). Similarly, for \( g^2(x) \), it transforms to \( 2g(x)g'(x) \). This use of the chain rule allows us to manage more complex differentiation cases effectively.

Sum Rule

The sum rule is one of the simpler rules for differentiation, making it easy to apply when dealing with sums of functions. The sum rule states that if you have two functions \( f(x) \) and \( g(x) \), the derivative of their sum \( (f(x) + g(x))' \) is simply the sum of their derivatives: \( f'(x) + g'(x) \). This rule is quite intuitive because it suggests that you can differentiate each part of the sum separately and then just add the results. In our specific exercise, the sum rule was used initially to separate the task of differentiating \( f^2(x) + g^2(x) \). Instead of directly dealing with the whole sum, we differentiated \( f^2(x) \) and \( g^2(x) \) individually. After using the chain rule to handle each derivative, we added the resulting derivatives together to get the final result. The sum rule is a straightforward but powerful rule, simplifying the differentiation process for expressions involving sums, such as our exercise scenario.