Implicit Differentiation and Derivatives

The video tutorial explains how to find the rate of change of y with respect to x for the equation y = cos(5x - 3y). It involves applying the derivative operator to both sides, using the chain rule, and solving for dy/dx through implicit differentiation. The process includes algebraic manipulation to isolate dy/dx, resulting in the final expression for the derivative.

What is the initial relationship given in the problem?

• y = tan(5x - 3y)

• y = 5x - 3y

• y = cos(5x - 3y)

• y = sin(5x - 3y)

What mathematical tool is applied to both sides of the equation to find the rate of change?

• Addition

• Derivative operator

• Multiplication

• Integration

Which rule is used to differentiate the cosine function in the equation?

• Chain rule

• Product rule

• Quotient rule

• Power rule

What is the derivative of 5x with respect to x?

• 3

• 5

• -3

• 0

What is the derivative of -3y with respect to x?

• -3

• 3

• 3 dy/dx

• -3 dy/dx

What algebraic method is used to solve for dy/dx after applying the chain rule?

• Factorization

• Distribution

• Elimination

• Substitution

What happens when you distribute the negative sine of (5x - 3y)?

• You get a tangent term

• You get a cosine term

• You get a negative sine term

• You get a positive sine term

What is the result of subtracting 3 sine of (5x - 3y) from both sides?

• It simplifies the equation

• It complicates the equation

• It eliminates dy/dx

• It adds a new term

What is the final expression for dy/dx?

• 3 times the sine of (5x - 3y)

• -5 times the sine of (5x - 3y) over 1 minus 3 sine of (5x - 3y)

• -3 times the sine of (5x - 3y)

• 5 times the sine of (5x - 3y)

What is the main challenge in solving implicit differentiation problems like this one?

• Solving for dy/dx

• Applying the chain rule

• Integrating the equation

• Finding the initial equation