Implicit Differentiation

What is implicit differentiation?

• A technique used to differentiate equations that define y explicitly in terms of x.

• A method for solving equations without any derivatives involved.

• A technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly solving for y.

• A process of finding the integral of a function without solving for y.

What is the slope of the tangent line to a curve at a point?

• The slope is the value of dy/dx at that point.

• The slope is always zero at any point on the curve.

• The slope is the average rate of change over the entire curve.

• The slope is determined by the second derivative of the function.

What does it mean if dy/dx is negative at a point?

• The function is increasing at that point.

• The function is constant at that point.

• The function is decreasing at that point.

• The function has a local maximum at that point.

How do you find dy/dx using implicit differentiation?

• Differentiate both sides of the equation with respect to x and solve for dy/dx.

• Integrate both sides of the equation with respect to x.

• Differentiate only the terms involving y and ignore x.

• Use the product rule for all terms in the equation.

What is the derivative of y^2 with respect to x?

• 2y(dy/dx)

• y^2

• 2y

• dy/dx

What is the derivative of x^2 with respect to x?

• 2x

• x^2

• 2

• x

What does it mean if dy/dx is positive at a point?

• The function is decreasing at that point.

• The function is constant at that point.

• The function is increasing at that point.

• The function has a local maximum at that point.

What is the derivative of a constant?

• 0

• 1

• The constant itself

• Undefined

How do you differentiate a product of two functions, such as xy?

• Use the sum rule: d(uv)/dx = u(dv/dx) + v(du/dx)

• Use the product rule: d(uv)/dx = u(dv/dx) + v(du/dx)

• Use the quotient rule: d(uv)/dx = (v(du/dx) - u(dv/dx))/v^2

• Use the chain rule: d(uv)/dx = du/dx * dv/dx

If x^2 + xy = 10, what is dy/dx when x = 2?

• 1

• -7/2

• 3

• 0