Implicit Differentiation and Tangent Line Slopes

The video tutorial explains how to find the slope of a tangent line using implicit differentiation. It starts by setting up the problem and calculating the y-value when x equals 1. The process of implicit differentiation is demonstrated step-by-step, leading to solving for dy/dx. Finally, the slope of the tangent line is calculated by substituting the values of x and y into the derived expression.

What is the main goal of the video tutorial?

• To calculate the area under a curve

• To determine the maximum value of a function

• To find the slope of a tangent line at a specific point

• To solve a quadratic equation

What is the y-value when x equals 1 in the given equation?

• 5

• 4

• 3

• 2

What is the first step in implicit differentiation?

• Add a constant to both sides

• Take the derivative of both sides with respect to x

• Divide both sides by x

• Multiply both sides by a constant

What is the derivative of a constant?

• One

• Zero

• The constant times x

• The constant itself

What expression do we solve for to find the slope of the tangent line?

• dx/dy

• dy/dx

• x/y

• y/x

What is the final expression for dy/dx before substituting values?

• 2x + 3y

• x^2 - 3y

• 3(y - x^2) - 2x

• 3x - 2y

What is the slope of the tangent line when x is 1 and y is 4?

• 1

• 25/27

• 27/25

• 0

What is the value of 3 times (4 - 1)^2?

• 18

• 36

• 9

• 27

What is the numerator of the slope expression after substitution?

• 25

• 31

• 29

• 27

What tool was used to generate the graph shown in the video?

• Desmos

• Wolfram Alpha

• GeoGebra

• Mathematica