A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100xy.

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Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so you'll need to use implicit differentiation.

Differentiate the left side: The derivative of $ \sqrt{3x^7} $ with respect to x is $ \frac{d}{dx}(\sqrt{3x^7}) $. Use the chain rule to find this derivative.

For the term $ y^2 $, apply the chain rule: $ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} $.

Differentiate the right side: For $ \sin^2(y) $, use the chain rule: $ \frac{d}{dx}(\sin^2(y)) = 2\sin(y)\cos(y) \frac{dy}{dx} $. For the term $ 100xy $, apply the product rule: $ \frac{d}{dx}(100xy) = 100(x \frac{dy}{dx} + y) $.

After differentiating, collect all terms involving $ \frac{dy}{dx} $ on one side of the equation and factor out $ \frac{dy}{dx} $. Solve for $ \frac{dy}{dx} $ to find the derivative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. In cases where y is defined implicitly by an equation involving both x and y, we differentiate both sides of the equation with respect to x, applying the chain rule to terms involving y. This allows us to find dy/dx without isolating y.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is crucial when differentiating terms involving y in implicit differentiation.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. In this context, recognizing that sin²y can be differentiated using the identity sin²y + cos²y = 1 is important. This understanding helps simplify the differentiation process and manage the terms effectively when applying implicit differentiation.

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