Differentiation and Implicit Functions

The video tutorial explains how to find the rate of change of x with respect to time (dx/dt) given a product of functions equation involving x and y. It starts by differentiating the equation using the product and chain rules, then isolates dx/dt. The tutorial simplifies the expression and calculates the exact and approximate values of dx/dt, using given values and solving for y.

What is the initial equation given in the problem?

• 2x + y^3 = 8

• 2xy^3 = 8

• x^2 + y^3 = 8

• 2x^3y = 8

Which rule is applied to differentiate the product of two functions?

• Power Rule

• Product Rule

• Chain Rule

• Quotient Rule

What is the derivative of y^3 with respect to t?

• 3y^2

• 3 dy/dt

• 3y^2 dy/dt

• y^3 dy/dt

What is the simplified form of the equation after applying the product rule?

• 2xy^3 dy/dt + 6y^2 dx/dt = 0

• 6xy^3 dy/dt + 2y^2 dx/dt = 0

• 6xy^2 dy/dt + 2y^3 dx/dt = 0

• 2xy^2 dy/dt + 6y^3 dx/dt = 0

How do you isolate dx/dt in the equation?

• Multiply both sides by 2y^3

• Subtract 6xy^2 dy/dt from both sides

• Divide both sides by 6xy^2

• Add 6xy^2 dy/dt to both sides

What is the value of y when x is substituted back into the original equation?

• y = 4

• y = cube root of 2

• y = 1

• y = 2

What operation is used to find y from y^3 = 2?

• Square both sides

• Subtract 2 from both sides

• Take the cube root of both sides

• Multiply both sides by 3

What is the final expression for dx/dt?

• 9 times the cube root of 2

• 18 divided by the cube root of 2

• 18 times the cube root of 2

• 9 divided by the cube root of 2

What is the approximate decimal value of dx/dt?

• 14.2866

• 18.2866

• 12.2866

• 16.2866

What is the value of dy/dt given in the problem?

• 2

• -2

• 3

• -3