The following equations implicitly define one or more functions.
a. Find dy/dx using implicit differentiation.
y² = x²(4 − x) / 4 + x (right strophoid)

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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 when differentiating terms involving y, use the chain rule.

Differentiate the left side: The derivative of y² with respect to x is 2y(dy/dx) using the chain rule.

Differentiate the right side: Use the quotient rule for the term x²(4 − x) / 4. The quotient rule states that if you have a function u/v, its derivative is (v(du/dx) - u(dv/dx)) / v².

Apply the product rule to differentiate x²(4 − x). The product rule states that if you have a function u(x)v(x), its derivative is u'(x)v(x) + u(x)v'(x).

After differentiating both sides, solve for dy/dx by isolating it on one side of the equation. This will give you the expression for dy/dx in terms of x and y.

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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. Instead of isolating y, we differentiate both sides of the equation with respect to x, applying the chain rule when necessary. This method is particularly useful for equations where y cannot be easily expressed as a function of x.

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05:14

Finding The Implicit Derivative

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 is the product of the derivative of y with respect to u and the derivative of u with respect to x. This is essential in implicit differentiation when differentiating terms involving y.

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Intro to the Chain Rule

Right Strophoid

The right strophoid is a specific type of curve defined by a particular mathematical equation. In the context of the given problem, it is represented by the equation y² = x²(4 − x) / 4 + x. Understanding the properties and shape of the right strophoid can provide insights into the behavior of the function and its derivatives, which is crucial for solving the implicit differentiation problem.

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