Use Equations 7 to find ∂z / ∂x and ∂z / ∂y e^z=x y z |...

Name: Problem 27, Chapter 11: Essential Calculus Early Transcendentals
Uploaded: 2020-05-25T03:27:36-08:00
Duration: 2 min 32 s
Channel: Issa Dababneh
Description: Use Equations 7 to find ∂z / ∂x and ∂z / ∂y e^z=x y z

Key Concepts

Product Rule

When differentiating expressions that are products of two or more functions, the product rule is applied. It states that the derivative of a product of functions is the derivative of the first function times the second plus the first times the derivative of the second. This is important when differentiating terms that involve the product of variables, such as in equations that mix multiplication of the variable and its derivative.

Chain Rule

The chain rule is used for differentiating composite functions. When a variable is defined in terms of another variable that is itself a function of another variable, the chain rule provides a method to compute the derivative by multiplying the derivative of the outer function by the derivative of the inner function, a key step in implicit differentiation for functions involving exponentials or other composite forms.

Implicit Differentiation

This concept deals with finding the derivative of a function that is defined implicitly rather than explicitly. In multivariable contexts, it involves differentiating both sides of an equation with respect to one variable while treating other variables as either functions or constants, which is essential when the dependent variable is intermingled with the independent variable in an equation.

Partial Derivatives

Partial derivatives are the derivatives of functions with several variables where we hold all other variables constant except the one of interest. They are pivotal in analyzing functions defined implicitly, as they allow us to examine the sensitivity of the function with respect to each independent variable separately.

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