Implicit Differentiation

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

• Differentiate both sides with respect to y, treating x as a function of y, yielding @@rac{dy}{dx} = rac{6y}{18x^2}@@.

• Differentiate both sides with respect to x, treating y as a function of x, yielding @@rac{dy}{dx} = rac{6x}{18y^2} = rac{x}{3y^2}@@.

• Differentiate both sides with respect to x, treating y as a constant, yielding @@rac{dy}{dx} = 6x - 18y^2@@.

• Differentiate both sides with respect to x, treating y as a function of x, yielding @@rac{dy}{dx} = rac{18y^2}{6x}@@.

What does the term 'secant line' refer to?

• A line that touches a curve at exactly one point.

• A line that intersects a curve at two or more points.

• A line that is parallel to the x-axis.

• A line that represents the maximum slope of a curve.

Tags

CCSS.HSF-IF.C.7D

What is the derivative of a function?

• The derivative is the area under the curve of the function.

• The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

• The derivative measures the total distance traveled by the function.

• The derivative is the maximum value of the function.

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.

• The derivative measures the area under the curve at a given point.

• The derivative is the average slope of the curve over an interval.

• The derivative indicates the maximum height of the curve at a specific point.

Tags

CCSS.HSF.IF.B.4

What is the slope of the tangent line?

• The slope of the tangent line is the second derivative of the function.

• The slope of the tangent line is given by the integral of the function.

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

• The slope of the tangent line is the average rate of change of the function over an interval.

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}@@.

• The chain rule states that the derivative of a product of functions is the product of their derivatives.

• The chain rule states that the derivative of a sum of functions is the sum of their derivatives.

• The chain rule states that the derivative of a function is always zero.

What is Implicit Differentiation?

• A method to differentiate equations where variables are separated.

• A technique to find the derivative of y with respect to x when y is defined implicitly as a function of x.

• A process to solve equations without derivatives.

• A way to integrate functions without using limits.

What is the quotient rule in differentiation?

• The derivative of a product of two functions is the product of their derivatives.

• 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}@@.

• The derivative of a sum of two functions is the sum of their derivatives.

• The derivative of a constant function is zero.

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}@@.

• The product rule states that the derivative of a product of two functions is the product of their derivatives.

• The product rule states that the derivative of a sum of two functions is the sum of their derivatives.

• The product rule states that the derivative of a function raised to a power is the power times the function raised to one less than that power.

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

• @@rac{dy}{dx} = -rac{F_x}{F_y}@@

• @@rac{dy}{dx} = rac{F_y}{F_x}@@

• @@rac{dy}{dx} = F_x + F_y@@

• @@rac{dy}{dx} = F_x - F_y@@