Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find all first partial derivatives of each function. \(f(s, t)=e^{t^{2}-s^{2}}\)

Step-by-Step Solution

Answer

The partial derivatives are \( \frac{\partial f}{\partial s} = -2s \cdot e^{t^2 - s^2} \) and \( \frac{\partial f}{\partial t} = 2t \cdot e^{t^2 - s^2} \).

Step 1: Identify the Variables

The function is given as \(f(s, t) = e^{t^2 - s^2}\). Here, \(s\) and \(t\) are the variables, and we need to find the partial derivatives with respect to each of these variables.

Step 2: Find the Partial Derivative with Respect to s

To find the partial derivative of \(f\) with respect to \(s\), we treat \(t\) as a constant and differentiate the function.\(\frac{\partial}{\partial s} f(s, t) = \frac{\partial}{\partial s} e^{t^2 - s^2}\)Using the chain rule, we differentiate the exponent \(t^2 - s^2\) with respect to \(s\):\( \frac{\partial}{\partial s} (t^2 - s^2) = -2s \).Thus, the partial derivative is:\(\frac{\partial f}{\partial s} = e^{t^2 - s^2} \cdot (-2s) = -2s \cdot e^{t^2 - s^2}. \)

Step 3: Find the Partial Derivative with Respect to t

Next, we find the partial derivative of \(f\) with respect to \(t\) by treating \(s\) as a constant and differentiating the function.\(\frac{\partial}{\partial t} f(s, t) = \frac{\partial}{\partial t} e^{t^2 - s^2}\)Using the chain rule, we differentiate the exponent \(t^2 - s^2\) with respect to \(t\):\( \frac{\partial}{\partial t} (t^2 - s^2) = 2t \).So, the partial derivative is:\(\frac{\partial f}{\partial t} = e^{t^2 - s^2} \cdot (2t) = 2t \cdot e^{t^2 - s^2}. \)

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used when differentiating composite functions. To understand the chain rule, consider a function composed of another function, such as \( f(g(x)) \). The chain rule allows us to differentiate such functions by taking into account the rates of change of both the inner function \( g(x) \) and the outer function \( f \).

For the function given in the exercise, \( f(s, t) = e^{t^2 - s^2} \), you see that the exponent \( t^2 - s^2 \) is itself a function of \( s \) and \( t \).
The chain rule helps to find how \( e^{t^2 - s^2} \) changes as \( s \) or \( t \) changes:

• When differentiating with respect to \( s \), you focus on \(-s^2\).
• When differentiating with respect to \( t \), \( t^2 \) becomes the focus.

The chain rule tells us to differentiate the outer function, \( e^u \) where \( u = t^2 - s^2 \), by its inner derivative, \( \frac{d}{ds}(-s^2) \) or \( \frac{d}{dt}(t^2) \).

This application significantly simplifies the process of finding partial derivatives in multivariable functions.

Differentiation

Differentiation is a process of calculating the rate of change of a function's output value with respect to its input. It's the pillar of calculus, allowing us to understand how functions transform as their variables shift.

When we differentiate a function, we're looking to find the slope or the rate at which the function changes at any given point. For the function \( f(s, t) = e^{t^2 - s^2} \), differentiation finds out how the expression changes as we tweak \( s \) or \( t \).

In the exercise:

• We treat all variables except one as constants to find each partial derivative.
• To differentiate with respect to \( s \), keep \( t \) constant and vice versa.

For example, to find \( \frac{\partial f}{\partial s} \), \( f \) is treated as a function of \( s \) alone, and \( t^2 \) becomes a constant.
This approach lets us zero in on how specifically each variable affects the outcome, making analysis of multivariable functions methodical and clear.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. It's an essential field for those studying sciences, engineering, economics, and many more areas where variables often intermingle.

The exercise requires finding the partial derivatives of \( f(s, t) = e^{t^2 - s^2} \), demonstrating core multivariable calculus techniques.

Important concepts include:

• Partial derivatives: Calculating the change in outputs by manipulating one variable at a time while holding others constant.
• The chain rule: Handling the differentiation of nested functions.

In multivariable calculus, these tools allow us to explore how fluctuations in each variable specifically influence the outcome.

This approach ensures detailed analysis and problem-solving in complex systems, giving insights that are not apparent when using single-variable calculus.