Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the first partial derivatives of the function. \(g(s, t)=s^{2} t+s t^{-3}\)

Step-by-Step Solution

Answer

The first partial derivatives of the function are: 1. \(\frac{\partial g(s, t)}{\partial s} = 2st + t^{-3}\) 2. \(\frac{\partial g(s, t)}{\partial t} = s^2 - 3st^{-4}\)

Step 1: Calculate the partial derivative with respect to s

To find the partial derivative of g(s,t) with respect to s, we'll treat t as a constant and use the power rule. The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Apply this rule to calculate the derivative of g(s,t) with respect to s: \(\frac{\partial g(s, t)}{\partial s} = \frac{\partial}{\partial s} (s^2 t + s t^{-3})\) \(\frac{\partial g(s, t)}{\partial s} = 2st + t^{-3}\)

Step 2: Calculate the partial derivative with respect to t

To find the partial derivative of g(s,t) with respect to t, we'll treat s as a constant and use the power rule. Apply this rule to calculate the derivative of g(s,t) with respect to t: \(\frac{\partial g(s, t)}{\partial t} = \frac{\partial}{\partial t} (s^2 t + s t^{-3})\) \(\frac{\partial g(s, t)}{\partial t} = s^2 - 3st^{-4}\) The first partial derivatives of the function are: 1. \(\frac{\partial g(s, t)}{\partial s} = 2st + t^{-3}\) 2. \(\frac{\partial g(s, t)}{\partial t} = s^2 - 3st^{-4}\)

Key Concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Fundamental to calculus is the concept of change and motion, characterizing how quantities evolve over time or in relation to each other.

It's split into two major parts: differential calculus, which deals with the rate of change of quantities, and integral calculus, which deals with the accumulation of quantities. When solving problems in calculus, such as finding the rate of change of a function at a point, we use derivatives, a measure that tells us how a function's value will change as its input changes.

Power Rule for Derivatives

The power rule is a basic technique applied in differential calculus for finding derivatives of functions that are powers of variables. It states that if you have a function of the form \(f(x) = x^n\) where \(n\) is any real number, the derivative of \(f\) with respect to \(x\) is \(f'(x) = nx^{n-1}\).

This rule significantly simplifies the process of differentiation by providing a quick and easy way to differentiate functions that would otherwise require a more tedious application of the limit definition of a derivative.

Multivariable Calculus

Multivariable calculus extends concepts from single-variable calculus to functions of multiple variables. This encompasses both partial derivatives and multiple integrals. It allows us to analyze functions that depend on several inputs and is incredibly useful for understanding phenomena across physics, engineering, and economics.

Here, rather than finding the derivative of a function with respect to a single variable, we investigate how the function changes in various directions, defined by different variables. This requires the use of partial derivatives, which measure a function's rate of change with respect to one variable while holding the others constant.

First Partial Derivatives

First partial derivatives are the fundamental tool in multivariable calculus for finding the rate of change of a function with respect to one variable at a time. When a function, like \(g(s, t)\), depends on more than one variable, the first partial derivative with respect to a particular variable treats all other variables as constants.

In the exercise, the first partial derivative of \(g(s, t)\) with respect to \(s\) treats \(t\) as constant and vice versa. Calculating these derivatives provides insight into how the function \(g\) changes in response to changes in each individual variable. This concept is critical in fields such as economics, physics, and engineering, where variables often influence each other.