Question

Given the equation y ^3 +xy=sin(x), share with the class on how to evaluate dy/dx ; assume that the equation implicitly defines y as a differentiable function of x.

a) Use the Chain Rule
b) Apply Implicit Differentiation
c) Implement the Power Rule
d) Utilize the Product Rule

Answer 1

In high school mathematics, to calculate the derivative dy/dx of y^3 + xy = sin(x), we apply rules such as the Chain Rule, Implicit Differentiation, Power Rule, and Product Rule, ultimately solving for dy/dx.

Step-by-step explanation:

The question involves calculating the derivative dy/dx of an implicitly defined function given by the equation y3 + xy = sin(x). To accomplish this, several calculus rules will be applied, including the Chain Rule, Implicit Differentiation, Power Rule, and Product Rule. Here's a step-by-step process:

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• Start by differentiating both sides of the equation with respect to x.
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• For the term y3, apply the Chain Rule and Power Rule: differentiate y3 as 3y2 and then multiply by dy/dx because y is a function of x.
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• For the term xy, apply the Product Rule: differentiate x with respect to x and y with respect to x separately, obtaining x(dy/dx) + y(1).
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• The right side of the equation, sin(x), simply differentiates to cos(x).
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• After differentiating, collect all terms containing dy/dx on one side and solve for dy/dx.

By carefully following these steps and using the rules of calculus, you can successfully differentiate the equation implicitly and find the derivative dy/dx.