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, without solving for y explicitly.

Front

How do you differentiate an equation implicitly?

Back

To differentiate an equation implicitly, apply the derivative to both sides of the equation, treating y as a function of x. Use the chain rule for terms involving y, which introduces dy/dx.

Front

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

Back

The derivative of y^n with respect to x is n*y^(n-1) * (dy/dx) using the chain rule.

Front

What is the chain rule in differentiation?

Back

The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In implicit differentiation, it applies when differentiating terms involving y.

Front

How do you find dy/dx for the equation x^3 + y^3 = 36?

Back

Differentiate both sides: 3x^2 + 3y^2(dy/dx) = 0. Solve for dy/dx to get dy/dx = -x^2/y^2.

Front

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

Back

Using the chain rule, the derivative is cos(y^2) * 2y * (dy/dx).

Front

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

Back

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

Front

What does it mean to find dy/dx at a given point?

Back

Finding dy/dx at a given point means evaluating the derivative at specific coordinates (x, y) to determine the slope of the tangent line at that point.

Front

How do you find dy/dx for the equation xy + y^2 = 2?

Back

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

Front

What is the significance of the negative sign in dy/dx = -y/(x + 2y)?

Back

The negative sign indicates that as x increases, y decreases, showing an inverse relationship between x and y in this context.