Find the indicated partial derivative. f(x, y)=y sin^-1(x y) ; fy(1, (1)/(2))

Name: Problem 42, Chapter 14: Calculus
Uploaded: 2023-11-29T07:00:04-08:00
Duration: 3 min 35 s
Channel: Melissa Munoz
Description: Find the indicated partial derivative. f(x, y)=y sin^-1(x y) ; fy(1, (1)/(2))

step 1 Consider the function f equal to y times the inverse sine of x times y. We want to find the part

Find the indicated partial derivative. f(x, y)=y sin^-1(x y)...

Key Concepts

Partial Derivatives

Partial derivatives measure the rate at which a multivariable function changes with respect to one variable while keeping other variables constant. This concept is essential in analyzing functions of several variables and is used to capture the sensitivity of the function along different coordinate directions.

Product Rule

The product rule is a fundamental differentiation technique used when differentiating a product of two functions. It states that the derivative of the product is the derivative of the first function times the second plus the first function times the derivative of the second. This concept is crucial when the function being differentiated is composed of multiple multiplicative factors.

Chain Rule

The chain rule is a differentiation method used for finding the derivative of composite functions. It helps in breaking down the process by first differentiating the outer function and then multiplying it by the derivative of the inner function. This rule is indispensable when dealing with functions that are applied within other functions.

Derivative of the Inverse Trigonometric Function

The derivative of an inverse trigonometric function, such as arcsin, is a key concept in calculus. In particular, the derivative of arcsin(u) with respect to u is 1/?(1 - u²). Understanding this derivative is important when differentiating functions that involve inverse trigonometric expressions, often in combination with the chain rule.

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