Understanding Derivatives of Parametric Functions

The video tutorial explains how to find the derivative of parametric functions, starting with basic examples and progressing to more complex ones involving trigonometric functions. It covers the process of finding dx/dt and dy/dt, and then using these to find dy/dx. The tutorial also includes examples with the variable theta and demonstrates the use of the chain rule in trigonometric contexts.

What is the first step in finding the derivative of a parametric function?

• Find the derivatives of x and y with respect to t

• Integrate the function with respect to t

• Find the second derivative of y with respect to x

• Find the derivative of x with respect to y

In the example where x = 8 + t^2 and y = 4t^2 - 5t^4, what is dy/dt?

• 4t^2 - 5t^4

• 8t - 20t^3

• 2t

• 8 + t^2

How do you find dy/dx from dy/dt and dx/dt?

• Multiply dy/dt by dx/dt

• Subtract dx/dt from dy/dt

• Divide dy/dt by dx/dt

• Add dy/dt and dx/dt

In the second example, what is the derivative of x = t^3 with respect to t?

• 3t^2

• t^2

• t^3

• 6t

What is the final expression for dy/dx in the second example?

• 6t^2 - 15t^3

• 4/t - 15

• 15t^2 - 4

• t^3 - 6t

In the trigonometric example with x = 4sin(t), what is dx/dt?

• 12cos(t)

• 12sin(t)

• 4sin(t)

• 4cos(t)

What is the simplified form of dy/dx for x = 4sin(t) and y = 12cos(t)?

• -3tan(t)

• 3tan(t)

• 3cot(t)

• -3cot(t)

In the example with x = 3sec(θ), what is the derivative of sec(θ) with respect to θ?

• sec(θ)tan(θ)

• tan(θ)

• cos(θ)

• sec^2(θ)

What is the final expression for dy/dx in the example with x = 3sec(θ) and y = 18tan(θ) - 5?

• 6cosec(θ)

• 6sec(θ)

• 6tan(θ)

• 6cot(θ)

In the final example, what rule is used to find the derivative of x = cos^3(θ)?

• Quotient rule

• Power rule

• Product rule

• Chain rule