Differentiation Techniques and Applications

This video tutorial covers implicit differentiation, starting with basic examples and moving to more complex applications involving the product rule and trigonometric functions. It also demonstrates how to find the second derivative using implicit differentiation. The tutorial includes step-by-step solutions to problems, emphasizing the importance of adding the term dydx when differentiating y variables with respect to x.

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

• 3y^2

• 3x^2 * dy/dx

• 3y^2 * dy/dx

• 3x^2

In the equation x^2 + y^2 = 100, what is the value of dy/dx?

• x/y

• y/x

• -x/y

• -y/x

What rule is used to differentiate the term 4xy in the equation x^3 + 4xy + y^2 = 13?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

What is the expression for dy/dx in the equation x^3 + 4xy + y^2 = 13?

• -(3x^2 - 4y)/(4x + 2y)

• -(3x^2 + 4y)/(4x + 2y)

• (3x^2 + 4y)/(4x + 2y)

• (3x^2 - 4y)/(4x + 2y)

What is the derivative of sin(xy^2) with respect to x?

• cos(xy^2) * (2y + x^2 * dy/dx)

• cos(xy^2)

• cos(xy^2) * (2xy + y^2 * dy/dx)

• cos(xy^2) * (y^2 + 2xy * dy/dx)

In the equation 5 - x^2 = sin(xy^2), what is the expression for dy/dx?

• (2x - y^2 * cos(xy^2))/(2xy * cos(xy^2))

• (-2x + y^2 * cos(xy^2))/(2xy * cos(xy^2))

• (2x + y^2 * cos(xy^2))/(2xy * cos(xy^2))

• (-2x - y^2 * cos(xy^2))/(2xy * cos(xy^2))

What is the first derivative dy/dx for the equation x^3 + y^3 = 9?

• -3x^2/3y^2

• x^2/y^2

• 3x^2/3y^2

• -x^2/y^2

Which rule is applied to find the second derivative of the function -x^2/y^2?

• Quotient Rule

• Product Rule

• Chain Rule

• Power Rule

What is the simplified form of the second derivative d^2y/dx^2 for the equation x^3 + y^3 = 9?

• -2x(y^3 + x^3)/y^5

• 2x(y^3 + x^3)/y^5

• -2x(y^3 - x^3)/y^5

• 2x(y^3 - x^3)/y^5

Evaluate the second derivative at the point (1, 2) for the equation x^3 + y^3 = 9.

• -9/16

• 9/16

• -16/9

• 16/9