Find d y / d x by implicit differentiation. y sin(x^2)=x sin(y^2)

Name: Problem 10, Chapter 2: Essential Calculus Early Transcendentals
Uploaded: 2020-05-26T20:03:36-08:00
Duration: 6 min 26 s
Channel: Will Cadelina
Description: Find d y / d x by implicit differentiation. y sin(x^2)=x sin(y^2)

step 1 To find D .Y over DX or the derivative of Y of this equation, we're going to start by differenti

Find d y / d x by implicit differentiation. y sin(x^2)=x...

Key Concepts

Product Rule

The product rule is a fundamental rule in differentiation used to find the derivative of the product of two functions. It states that the derivative of a product is equal to the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function. This principle is essential when differentiating expressions where both factors depend on the independent variable.

Chain Rule

The chain rule is a critical tool in differentiation for handling composite functions, where one function is applied to the result of another function. It provides a method for computing the derivative by taking the derivative of the outer function evaluated at the inner function and then multiplying it by the derivative of the inner function. This rule is indispensable when the arguments of functions are themselves functions of another variable.

Implicit Differentiation

Implicit differentiation is a technique used when a function is not explicitly solved for one variable in terms of another. Instead of rewriting the equation to have one variable isolated, both sides of the equation are differentiated with respect to the independent variable, treating the dependent variable as an implicit function. This method is especially useful when the relationship between the variables is complicated or intertwined.

Differentiation of Trigonometric Functions

Differentiating trigonometric functions involves using the standard derivatives of sine, cosine, and other trigonometric functions. When these functions have arguments that are themselves functions of the independent variable, the chain rule is applied to properly account for the rate of change of the inner functions. This concept ensures accurate computation of derivatives in contexts involving trigonometric expressions.

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