Implicit Differentiation

Front

What is Implicit Differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both dependent and independent variables, without solving for one variable in terms of the other.

Front

How do you apply implicit differentiation to the equation x + 4y = 1?

Back

Differentiate both sides with respect to x: 1 + 4(dy/dx) = 0. Solve for dy/dx to get dy/dx = -1/4.

Front

What is the derivative of y with respect to x when given the equation 9y + x = 4y?

Back

Rearranging gives 5y = -x, so dy/dx = -1/5.

Front

How do you find dy/dx for the equation xy = 6?

Back

Differentiate both sides: y + x(dy/dx) = 0. Thus, dy/dx = -y/x.

Front

What is the derivative of y when given the equation xy + y^2 = 2?

Back

Differentiate: y + x(dy/dx) + 2y(dy/dx) = 0. Solve to get dy/dx = -y/(x + 2y).

Front

How do you find dy/dx at a specific point, e.g., x^3 + 2xy - y^2 = 11 at (2,3)?

Back

Differentiate implicitly, substitute x=2 and y=3 into the derivative to find dy/dx.

Front

What is the chain rule in implicit differentiation?

Back

The chain rule states that if y is a function of u, and u is a function of x, then dy/dx = (dy/du)(du/dx). This is used when differentiating y implicitly.

Front

What is the product rule in implicit differentiation?

Back

The product rule states that if u and v are functions of x, then d(uv)/dx = u(dv/dx) + v(du/dx). This is applied when differentiating products of functions.

Front

What is the quotient rule in implicit differentiation?

Back

The quotient rule states that if u and v are functions of x, then d(u/v)/dx = (v(du/dx) - u(dv/dx))/v^2. This is used for differentiating quotients of functions.

Front

What is the significance of dy/dx in a function?

Back

dy/dx represents the slope of the tangent line to the curve at a given point, indicating how y changes with respect to x.