AP Calculus 4.3 Implicit Differentiation

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What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both dependent and independent variables.

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How do you apply implicit differentiation to an equation?

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To apply implicit differentiation, differentiate both sides of the equation with respect to the independent variable, treating the dependent variable as a function of the independent variable, and apply the chain rule where necessary.

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What is the chain rule in differentiation?

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The chain rule states that if a function y is defined as a function of u, which is in turn 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: dy/dx = (dy/du) * (du/dx).

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What is the derivative of y with respect to x when using implicit differentiation?

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When using implicit differentiation, the derivative of y with respect to x is denoted as dy/dx, and it is found by isolating dy/dx in the differentiated equation.

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What is the first step in implicit differentiation?

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The first step in implicit differentiation is to differentiate both sides of the equation with respect to the independent variable, usually x.

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How do you differentiate a product of two functions in implicit differentiation?

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To differentiate a product of two functions, use the product rule: if u and v are functions of x, then d(uv)/dx = u(dv/dx) + v(du/dx).

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What is the product rule in differentiation?

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The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

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What is the quotient rule in differentiation?

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The quotient rule states that if you have a function that is the quotient of two functions, u and v, then the derivative is given by (v(du/dx) - u(dv/dx)) / v^2.

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How do you find dy/dx from an implicit equation?

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To find dy/dx from an implicit equation, differentiate both sides with respect to x, collect all terms involving dy/dx on one side, and then solve for dy/dx.

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What is the significance of the implicit function theorem?

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The implicit function theorem provides conditions under which a relation defines a function implicitly, allowing for the differentiation of the function even when it is not explicitly solved for one variable.