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 x and y, rather than explicitly solving for y.

Front

How do you apply implicit differentiation to an equation?

Back

To apply implicit differentiation, differentiate both sides of the equation with respect to x, treating y as a function of x. Use the chain rule for terms involving y.

Front

What is the derivative of y with respect to x denoted as?

Back

The derivative of y with respect to x is denoted as dy/dx.

Front

What is the chain rule in the context of implicit differentiation?

Back

The chain rule states that if y is a function of x, then the derivative of y with respect to x is the derivative of y with respect to y times the derivative of y with respect to x.

Front

Given the equation xy + y^2 = 2, how do you find dy/dx?

Back

Differentiate both sides with respect to x: d(xy)/dx + d(y^2)/dx = d(2)/dx. Apply the product rule for xy and the chain rule for y^2.

Front

What is the product rule?

Back

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

Front

How do you differentiate y^2 with respect to x?

Back

Using the chain rule, d(y^2)/dx = 2y(dy/dx).

Front

What is the implicit differentiation of the equation 2x^4y^2 - y^3 = 7?

Back

Differentiate both sides: 8x^3y^2 + 4x^4(2y(dy/dx)) - 3y^2(dy/dx) = 0, then solve for dy/dx.

Front

What is the derivative of the equation x^2 + xy + y^3 = 0?

Back

Differentiate: 2x + (x(dy/dx) + y) + 3y^2(dy/dx) = 0, then solve for dy/dx.

Front

How do you find dy/dx for the equation 2x^7y^4 = 10?

Back

Differentiate both sides: 14x^6y^4 + 8x^7y^3(dy/dx) = 0, then solve for dy/dx.