Implicit Differentiation and Trigonometric Identities

Professor Dave explains implicit differentiation, a technique used to differentiate functions not explicitly solved for one variable in terms of another. He demonstrates this with examples, including a circle equation and more complex functions involving product and chain rules. The video concludes with a summary and comprehension check.

Why is implicit differentiation necessary for some functions?

• It simplifies the function to a constant.

• It only applies to linear functions.

• It allows differentiation without solving for one variable in terms of another.

• It eliminates the need for the chain rule.

What is the first step in implicit differentiation?

• Solve for y in terms of x.

• Differentiate both sides with respect to x.

• Apply the product rule.

• Set the equation equal to zero.

In the equation x^2 + y^2 = 9, what is the derivative of y^2 with respect to x?

• 0

• 2x

• 2y

• 2y * dy/dx

When applying the product rule in implicit differentiation, what must you remember?

• Only differentiate the first term.

• Ignore the chain rule.

• Set the derivative equal to zero.

• Differentiate each part of the product separately.

How do you handle the derivative of y when using implicit differentiation?

• Use the chain rule to include dy/dx.

• Treat y as a constant.

• Ignore y completely.

• Differentiate y with respect to y.

In the equation x^3 + (x^2 * y) + 4y^2 = 6, what rule is used for the term (x^2 * y)?

• Chain rule

• Power rule

• Product rule

• Quotient rule

What is the derivative of y^2 with respect to x in the equation (x^2 * y) + (x * y^2) = 3x?

• 2y

• 2y * dy/dx

• 2x

• 0

What is the result of differentiating 4cos(x)sin(y) with respect to x?

• -4sin(x)sin(y)

• 4sin(x)cos(y)

• 4cos(x)cos(y) * dy/dx

• -4sin(x)cos(y) * dy/dx

What trigonometric identity is used to simplify the result of the implicit differentiation in the final example?

• Sine rule

• Cosine rule

• Tangent identity

• Pythagorean identity

What is the final expression for y prime in the equation 4cos(x)sin(y) = 1?

• tan(x)cos(y)

• cos(x)cos(y)

• sin(x)sin(y)

• tan(x)tan(y)