Problem 11 - Calculus for Biology and Medicine

Question

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=e^{x} \sin (x y)\)

Step-by-Step Solution

Answer

\(\frac{\partial f}{\partial x} = e^x \sin(xy) + e^x y \cos(xy)\) and \(\frac{\partial f}{\partial y} = e^x x \cos(xy)\).

Step 1: Understand the Problem

We need to find the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) for the function \(f(x, y) = e^x \sin(xy)\). This involves treating each variable independently while taking the derivative.

Step 2: Find \(\frac{\partial f}{\partial x}\)

To find the partial derivative with respect to \(x\), differentiate \(f(x, y) = e^x \sin(xy)\) with respect to \(x\) while treating \(y\) as a constant. Use the product rule: \((uv)' = u'v + uv'\), where \(u = e^x\) and \(v = \sin(xy)\). So, \(u' = e^x\) and \(v' = y\cos(xy)\).Apply the product rule:\[\frac{\partial f}{\partial x} = e^x \cdot \sin(xy) + e^x y \cos(xy)\]

Step 3: Find \(\frac{\partial f}{\partial y}\)

To find the partial derivative with respect to \(y\), differentiate \(f(x, y) = e^x \sin(xy)\) with respect to \(y\) while treating \(x\) as a constant. The derivative of \(\sin(xy)\) with respect to \(y\) is \(x \cos(xy)\).Hence:\[\frac{\partial f}{\partial y} = e^x \cdot x \cos(xy)\]

Step 4: Write the Results

Summarize the derivatives:The partial derivative of \(f\) with respect to \(x\) is:\[\frac{\partial f}{\partial x} = e^x \sin(xy) + e^x y \cos(xy)\]The partial derivative of \(f\) with respect to \(y\) is:\[\frac{\partial f}{\partial y} = e^x x \cos(xy)\]

Key Concepts

Partial Derivatives

When working with functions of multiple variables, such as the given function \( f(x, y) = e^x \sin(xy) \), we often need to explore how changes in one variable affect the function while keeping the other variable constant. This exploration leads us to partial derivatives.

A partial derivative of a function is essentially the derivative of the function with respect to one of its variables, treating all other variables as constants. For example:

• The partial derivative \( \frac{\partial f}{\partial x} \) provides the rate of change of \( f \) when \( x \) changes and \( y \) remains constant.
• The partial derivative \( \frac{\partial f}{\partial y} \) does the same but for changes in \( y \) while \( x \) stays constant.

By understanding partial derivatives, we can better analyze and model real-world problems involving more than one variable, such as in engineering or physics.

Product Rule

In calculus, the product rule is a key method for finding the derivative of a product of two functions. It states that if we have two differentiable functions \( u(x) \) and \( v(x) \), the derivative of their product \( u(x)v(x) \) with respect to \( x \) is given by \[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]

When we apply the product rule in the context of partial derivatives, we treat one variable as the primary variable and others as constants. In our exercise, for \( f(x, y) = e^x \sin(xy) \), we identified the parts of the function as:

• \( u = e^x \) and \( v = \sin(xy) \)
• \( u' = e^x \) and \( v' = y\cos(xy) \)

Applying the product rule helped us find that:

• \( \frac{\partial f}{\partial x} = e^x \sin(xy) + e^x y \cos(xy) \)

This tool is particularly useful when dealing with complex expressions involving products, making the process of finding derivatives manageable.

Trigonometric Functions

Trigonometric functions, like \( \sin \) and \( \cos \), often appear in calculus problems involving oscillatory or periodic behavior. In the given function, \( f(x, y) = e^x \sin(xy) \), \( \sin(xy) \) is the trigonometric component.

Trigonometric functions have well-known derivatives:

• The derivative of \( \sin(x) \) is \( \cos(x) \)
• The derivative of \( \cos(x) \) is \(-\sin(x) \)

In partial derivatives, you apply these rules by differentiating with respect to one variable while holding others constant. For example, in \( \frac{\partial f}{\partial y} \), the derivative of \( \sin(xy) \) with respect to \( y \) translates to :

• \( x \cos(xy) \).

These rules simplify the process of working with trigonometric expressions, helping to unravel their role in more complex functions. Recognizing their patterns quickly becomes a vital skill in calculus.