Calculating Derivatives and Rules

The video tutorial explains how to find the derivative of a vector valued function r(t) with components x(t), y(t), and z(t). It covers the differentiation of each component using the chain rule for x(t) and the quotient rule for y(t) and z(t). The tutorial provides step-by-step instructions and simplifications to arrive at the derivatives of each component, resulting in the complete derivative r'(t).

What are the components of the vector valued function r(t)?

• a(t), b(t), c(t)

• p(t), q(t), r(t)

• x(t), y(t), z(t)

• m(t), n(t), o(t)

Which rule is used to find the derivative of the x component?

• Power Rule

• Product Rule

• Quotient Rule

• Chain Rule

What is the derivative of x(t) when x(t) = 3 / (-5t - 7)?

• -3 / (-5t - 7)^2

• -15 / (-5t - 7)^2

• 15 / (-5t - 7)^2

• 3 / (-5t - 7)^2

What is the purpose of using the chain rule in the context of this video?

• To find the derivative of a quotient

• To find the derivative of a sum

• To find the derivative of a product

• To find the derivative of a composite function

Which rule is applied to find the derivative of the y component?

• Sum Rule

• Product Rule

• Quotient Rule

• Chain Rule

What is the derivative of y(t) when y(t) = -2t / (3t^2 + 4)?

• (6t^2 + 8) / (3t^2 + 4)^2

• (-6t^2 + 8) / (3t^2 + 4)^2

• (6t^2 - 8) / (3t^2 + 4)^2

• (-6t^2 - 8) / (3t^2 + 4)^2

Which mathematical operation is used to simplify the expression for y'(t)?

• Subtraction

• Division

• Addition

• Multiplication

What is the initial step in applying the quotient rule to find z'(t)?

• Differentiate the numerator

• Square the denominator

• Multiply the numerator by the derivative of the denominator

• Add the numerator and denominator

What is the derivative of z(t) when z(t) = -4t^2 / (-t^3 - 7)?

• (4t^4 - 56t) / (-t^3 - 7)^2

• (-4t^4 + 56t) / (-t^3 - 7)^2

• (-4t^4 - 56t) / (-t^3 - 7)^2

• (4t^4 + 56t) / (-t^3 - 7)^2

What is the final expression for r'(t) after finding all components?

• r'(t) = (15 / (-5t - 7)^2, (6t^2 - 8) / (3t^2 + 4)^2, (-4t^4 + 56t) / (-t^3 - 7)^2)

• r'(t) = (15 / (-5t - 7)^2, (6t^2 + 8) / (3t^2 + 4)^2, (4t^4 + 56t) / (-t^3 - 7)^2)

• r'(t) = (15 / (-5t - 7)^2, (-6t^2 + 8) / (3t^2 + 4)^2, (4t^4 - 56t) / (-t^3 - 7)^2)

• r'(t) = (-15 / (-5t - 7)^2, (6t^2 + 8) / (3t^2 + 4)^2, (-4t^4 - 56t) / (-t^3 - 7)^2)