Problem 47 - Calculus for Biology and Medicine

Question

In Problems \(39-48\), find the indicated partial derivatives. $$ f(x, y)=\ln (x+y) ; \frac{\partial^{3} f}{\partial x^{3}} $$

Step-by-Step Solution

Answer

\( \frac{2}{(x+y)^3} \)

Step 1: Understand the Problem

We need to find the third partial derivative of the function \( f(x, y) = \ln(x + y) \) with respect to \( x \). This involves differentiating the function three times with respect to \( x \), while treating \( y \) as a constant.

Step 2: Compute the First Partial Derivative with Respect to x

The first partial derivative of \( f(x, y) = \ln(x+y) \) with respect to \( x \) is found by applying the chain rule. Since \( y \) is treated as constant, we have: \[ \frac{\partial f}{\partial x} = \frac{1}{x+y} \]

Step 3: Compute the Second Partial Derivative with Respect to x

Now, we differentiate the first partial derivative \( \frac{\partial f}{\partial x} = \frac{1}{x+y} \) again with respect to \( x \). This requires applying the derivative of \( \frac{1}{u} \), which is \( -\frac{1}{u^2} \cdot \frac{du}{dx} \). Thus, we get: \[ \frac{\partial^2 f}{\partial x^2} = -\frac{1}{(x+y)^2} \]

Step 4: Compute the Third Partial Derivative with Respect to x

Finally, we differentiate the second partial derivative \( \frac{\partial^2 f}{\partial x^2} = -\frac{1}{(x+y)^2} \) one more time with respect to \( x \). Using the same technique: \[ \frac{\partial^3 f}{\partial x^3} = 2\frac{1}{(x+y)^3} \]

Step 5: Conclusion

The third partial derivative of \( f(x, y) = \ln(x+y) \) with respect to \( x \) is \( \frac{2}{(x+y)^3} \).

Key Concepts

The Chain Rule

The chain rule is a fundamental concept in calculus that helps us differentiate compositions of functions. In a multivariable context, it's especially useful when dealing with functions of more than one variable. For instance, when we examine the function \(f(x, y) = \ln(x + y)\), we see that it's a composite function of \(x\) and \(y\).
To apply the chain rule, we treat one variable as a function of another while differentiating. This approach is critical when finding partial derivatives, especially with functions like \(f(x, y)\), where each variable plays a distinct yet interconnected role. Let's break it down:

• First, we differentiate with respect to \(x\), treating \(y\) as a constant. The derivative of \(\ln(u)\), where \(u = x + y\), is \(\frac{1}{u}\). Therefore, the first partial derivative is \(\frac{1}{x+y}\).
• As we proceed to higher derivatives, we use the chain rule again, considering the effects of our choice of variable.

This cascading application of the chain rule ensures we accurately capture changes as one variable shifts while holding others constant.

Understanding Higher Order Derivatives

Higher order derivatives are derivatives of derivatives, indicating how rates of change themselves change. For instance, taking the third partial derivative with respect to \(x\) reveals insights into the curvature and behavior of the function \(f(x, y) = \ln(x+y)\) across the \(x\)-axis.
By computing higher order derivatives, you can:

• Analyze the concavity of a function by observing the sign and value of the second derivative. In our example, \(\frac{\partial^2 f}{\partial x^2} = -\frac{1}{(x+y)^2}\) tells us how the function's slope behaves.
• Understand even deeper behavior with the third partial derivative. For \(\frac{\partial^3 f}{\partial x^3} = \frac{2}{(x+y)^3}\), this tells us about the rate of change of the concavity itself, providing insights into how rapidly the curve is altering its slope direction.

These insights are crucial for advanced calculus problems, offering a deeper look into the structure of multivariable functions.

Exploring Multivariable Functions

Multivariable functions involve more than one independent variable, offering a rich canvas for calculus exploration. A function like \(f(x, y) = \ln(x+y)\) depends on two variables, requiring specific techniques to analyze changes in output when these inputs vary.
Working with multivariable functions involves:

• Understanding partial derivatives, which measure how the function changes in response to one variable while keeping others constant.
• Applying concepts like the gradient, which is a vector showing the direction of the steepest ascent of function values.
• Using tools like the partial derivative to gain insights into the overall behavior of the function by observing dependence on separate variables like \(x\) and \(y\).

When graphed, multivariable functions often produce surfaces. Differentiating them helps uncover how these surfaces bend and twist across different planes, an essential skill in fields such as physics, engineering, and economics.