How do you apply implicit differentiation to the equation $$3x^2 - 6y^3 = 13$$ ?

Differentiate both sides with respect to x, treating y as a function of x. This gives $$6x - 18y^2 \frac{dy}{dx} = 0$$ . Rearranging yields $$\frac{dy}{dx} = \frac{6x}{18y^2} = \frac{x}{3y^2}$$ .

Differentiate the equation $$x^3 + y^3 = 36$$ using implicit differentiation.

Differentiating gives $$3x^2 + 3y^2 \frac{dy}{dx} = 0$$ . Solving for $$\frac{dy}{dx}$$ yields $$\frac{dy}{dx} = -\frac{x^2}{y^2}$$ .

What is the derivative of $$9y + x = 4y$$ ?

Rearranging gives $$9y - 4y + x = 0$$ . Differentiating gives $$5 \frac{dy}{dx} + 1 = 0$$ , leading to $$\frac{dy}{dx} = -\frac{1}{5}$$ .

How do you differentiate the equation $$-4x^2 + 12x + 6y^2 - 10y = 5$$ ?

Differentiate both sides: $$-8x + 12 + 12y \frac{dy}{dx} - 10 \frac{dy}{dx} = 0$$ . Rearranging gives $$\frac{dy}{dx} = \frac{8x - 12}{12y - 10}$$ .

What is the slope of the tangent line?

The slope of the tangent line at a point on a curve is given by the derivative of the function at that point.

What does the term 'secant line' refer to?

A secant line is a line that intersects a curve at two or more points. The slope of the secant line between two points on the curve approximates the average rate of change of the function between those points.

What is the relationship between the derivative and the slope of a curve?

The derivative at a point gives the slope of the tangent line to the curve at that point, indicating how steep the curve is at that location.

What is the significance of the derivative being zero?

If the derivative of a function is zero at a point, it indicates that the function has a horizontal tangent line at that point, which may correspond to a local maximum, minimum, or inflection point.

What is the chain rule in differentiation?

The chain rule states that if a variable y depends on u, which in turn depends on x, then the derivative of y with respect to x is given by $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$ .

What is the product rule in differentiation?

The product rule states that if two functions u(x) and v(x) are multiplied, then the derivative is given by $$\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$$ .

What is the quotient rule in differentiation?

The quotient rule states that if a function is the quotient of two functions u(x) and v(x), then the derivative is given by $$\frac{d(\frac{u}{v})}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$ .

What is the importance of implicit differentiation in calculus?

Implicit differentiation is important because it allows us to differentiate equations that cannot be easily solved for one variable in terms of another, enabling us to find derivatives of more complex relationships.

Implicit Differentiation Flashcards

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1.

FLASHCARD QUESTION

Front

What is Implicit Differentiation?

Back

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not separated. It allows us to find the derivative of y with respect to x when y is defined implicitly as a function of x.

2.

FLASHCARD QUESTION

Front

What is the derivative of a function?

Back

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. It represents the rate of change of the function with respect to its variable.

3.

FLASHCARD QUESTION

Front

Back

4.

FLASHCARD QUESTION

Front

What is the formula for the derivative of y with respect to x in implicit differentiation?

Back

5.

FLASHCARD QUESTION

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Back

6.

FLASHCARD QUESTION

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7.

FLASHCARD QUESTION

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