In Exercises 1-22, find ∂f / ∂x and ∂f / ∂y f(x, y)=e^x y lny

Name: Problem 16, Chapter 14: Thomas Calculus
Uploaded: 2019-06-12T22:28:26-08:00
Duration: 1 min 48 s
Channel: Brittany Knowlton
Description: In Exercises 1-22, find ∂f / ∂x and ∂f / ∂y f(x, y)=e^x y lny

step 1 So I have the function f of xy is equal to e to the xy times natural log of y, and I want to fin

In Exercises 1-22, find ∂f / ∂x and ∂f / ∂y f(x, y)=e^x y...

Key Concepts

Partial Derivative

A partial derivative measures how a multivariable function changes as one of its variables changes while keeping the other variables constant. It is a fundamental concept in multivariable calculus that extends the idea of the derivative from functions of a single variable to functions of several variables.

Product Rule

The product rule is used when differentiating the product of two functions. In the context of multivariable functions, when one component depends on the differentiation variable and the other does not, the rule helps combine their derivatives appropriately.

Chain Rule

The chain rule is essential for differentiating composite functions. It allows one to take into account the inner function’s derivative when a function is composed with another function, which is particularly useful when the argument of an exponential function is a product of variables.

Differentiation of Exponential Functions

Differentiating an exponential function involves applying the exponential function's unique property where the derivative of e^(g(x)) is e^(g(x)) multiplied by the derivative of the exponent g(x). This concept is frequently used when dealing with functions where the exponent itself is a function of the variable.

Differentiation of Logarithmic Functions

The differentiation of a natural logarithm involves applying the rule that the derivative of ln(x) is 1/x. In the context of functions of several variables, when one variable is present inside the logarithm, it may be treated as a constant with respect to other variables during partial differentiation.

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