Problem 39 - Calculus for Biology and Medicine

Question

Find the indicated partial derivatives. \(f(x, y)=x^{2} y^{2}+2 x y^{2} ; \frac{a^{2} f}{\partial x^{2}}\)

Step-by-Step Solution

Answer

The second partial derivative \(\frac{\partial^2 f}{\partial x^2}\) is \(2y^2\).

Step 1: Understand the Function

We are given the function \( f(x, y) = x^2 y^2 + 2xy^2 \). The function is defined with two variables: \(x\) and \(y\).

Step 2: Differentiate with Respect to x (First Derivative)

To find the partial derivative of \(f\) with respect to \(x\), consider \(y\) as a constant. Differentiate each term separately: - The derivative of \(x^2 y^2\) with respect to \(x\) is \(2xy^2\).- The derivative of \(2xy^2\) with respect to \(x\) is \(2y^2\).The first derivative \(\frac{\partial f}{\partial x}\) is: \[\frac{\partial f}{\partial x} = 2xy^2 + 2y^2\].

Step 3: Differentiate Again with Respect to x (Second Derivative)

Now take the derivative of \(\frac{\partial f}{\partial x} = 2xy^2 + 2y^2\) with respect to \(x\), again treating \(y\) as a constant:- The derivative of \(2xy^2\) with respect to \(x\) is \(2y^2\).- The derivative of \(2y^2\) with respect to \(x\) is \(0\), since it is a constant in terms of \(x\).The second derivative, \(\frac{\partial^2 f}{\partial x^2}\), is:\[\frac{\partial^2 f}{\partial x^2} = 2y^2\].

Key Concepts

Multivariable Calculus

Multivariable Calculus is an extension of traditional calculus, where we study functions with more than one variable. In this context, functions can be defined in terms of more than one independent variable, like in our example function:

\[ f(x, y) = x^2 y^2 + 2xy^2 \]

Here, both \(x\) and \(y\) are variables that can change independently. Multivariable calculus focuses on understanding how changes in these variables affect the function's overall behavior.

• Partial Derivatives: These help us measure how a function changes as one specific variable changes, while keeping others constant.
• Applications: Useful in fields like engineering and physics to model complex systems where multiple factors interact.

By considering the effects of each variable separately, multivariable calculus provides a comprehensive framework for analyzing and optimizing real-world phenomena.

Differentiation

Differentiation is a key concept in calculus that deals with finding the rate of change of a function. When applied to multivariable calculus, we use partial differentiation, where we take the derivative with respect to one variable while treating others as constants.

For the given function \(f(x, y)\), we analyzed how \(f\) changes concerning \(x\), leading to the first partial derivative \(\frac{\partial f}{\partial x}\). This step involves straightforward computation of derivatives:

• First Derivative: Differentiating \(x^2 y^2\) gives \(2xy^2\), treating \(y^2\) as a constant.
• First Derivative: Differentiating \(2xy^2\) yields \(2y^2\), again considering \(y^2\) constant.

The resulting derivative helps identify the function's instantaneous rate of change in the \(x\) direction.

Partial derivatives play a crucial role in studying surfaces, gradients, and optimization problems in multiple dimensions.

Second Derivative

The second derivative is a further step in understanding the behavior of a function, providing insights into its concavity and rate of variation changes. For functions involving multiple variables, the second partial derivative can be a bit more nuanced.

In our task, after finding the first derivative, we proceeded to find the second derivative \(\frac{\partial^2 f}{\partial x^2}\), which tells us about the curvature of the function relating \(x\).

• The Second Derivative of \(2xy^2\): Differentiating once more with respect to \(x\) gives \(2y^2\), highlighting constant slope changes as \(y^2\) remains unaltered.
• The Constant Term \(2y^2\): Its derivative remains \(0\) since it does not vary with \(x\).

Understanding second derivatives in multivariable calculus is vital for assessing how sharply a function changes direction. This information is essential for more advanced applications, such as finding critical points and optimizing functions across multiple dimensions.