Implicit Differentiation

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What is implicit differentiation?

Back

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly as y = f(x). It involves differentiating both sides of the equation with respect to x and applying the chain rule.

Front

How do you find the derivative of an implicit function?

Back

To find the derivative of an implicit function, differentiate both sides of the equation with respect to x, treating y as a function of x. Then, solve for dy/dx.

Front

What is the formula for the slope of the tangent line at a point (x_0, y_0)?

Back

The slope of the tangent line at a point (x_0, y_0) is given by the derivative dy/dx evaluated at that point.

Front

How do you write the equation of a tangent line?

Back

The equation of a tangent line can be written in point-slope form: y - y_0 = m(x - x_0), where m is the slope at the point (x_0, y_0).

Front

What is the chain rule in differentiation?

Back

The chain rule states that if a function y is defined implicitly in terms of x, then the derivative dy/dx can be found using dy/dx = dy/du * du/dx, where u is an intermediate variable.

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Back

Using implicit differentiation, we get: 2x + y + x(dy/dx) + 2y(dy/dx) = 0, which simplifies to dy/dx = -\frac{2x + y}{x + 2y}.

Front

What is the significance of the derivative in the context of curves?

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The derivative represents the rate of change of y with respect to x, indicating the slope of the tangent line to the curve at any given point.

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Back

Differentiate both sides: 2y^3 + 6xy^2(dy/dx) - 2xy - x^2(dy/dx) = 0. Solve for dy/dx to get dy/dx = \frac{2y(x - y^2)}{x(6y^2 - x)}.

Front

What is the implicit function theorem?

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The implicit function theorem states that if a function is defined implicitly by an equation and certain conditions are met, then it is possible to express one variable as a function of the other.

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Back

Using implicit differentiation, we get: -3\cos x - 4\sin y(dy/dx) = 0, leading to dy/dx = -\frac{3\cos x}{4\sin y}.