Understanding Differentiable Functions and Chain Rule

The video tutorial explains how differentiable functions x and y are related by the equation y = √x. It discusses how x and y are functions of t and uses the chain rule to find the derivative of y with respect to t when x = 9 and dx/dt = 12. The solution involves applying the power rule and simplifying the expression to find that the derivative of y with respect to t is 2.

What is the relationship between the functions x and y in the given problem?

• y = x^2

• y = √x

• y = x + 1

• y = 2x

What is the derivative of x with respect to t as given in the problem?

• 10

• 12

• 8

• 15

How can y be considered a function of t?

• Because y is independent of x

• Because y is a constant

• Because y is a function of x, which is a function of t

• Because y is directly a function of t

What mathematical rule is applied to find the derivative of y with respect to t?

• Chain Rule

• Product Rule

• Quotient Rule

• Sum Rule

What is the expression for the derivative of y with respect to x using the power rule?

• x^2

• 2x

• 1/2 x^(-1/2)

• x^(-1/2)

What is the value of x when the derivative of y with respect to t is calculated?

• 1

• 4

• 9

• 16

What is the final value of the derivative of y with respect to t when x = 9?

• 3

• 1

• 4

• 2

What is the value of 9 raised to the power of -1/2?

• 9

• 1/3

• 3

• 1/9

What is the result of multiplying 1/2 by 1/3?

• 1/6

• 1/3

• 1/2

• 1/9

What is the final step in calculating the derivative of y with respect to t?

• Adding the derivatives

• Multiplying by the derivative of x with respect to t

• Subtracting the derivatives

• Dividing by the derivative of x with respect to t