Implicit Differentiation Practice

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 differentiate y = sin(2x^2)?

Back

Using the chain rule: y' = cos(2x^2) * d(2x^2)/dx = cos(2x^2) * 4x = 4xcos(2x^2).

Front

What is the derivative of xy + y^2 = 2 with respect to x?

Back

Using implicit differentiation: dy/dx = -y/(x + 2y).

Front

How do you find dy/dx for the equation 2xy^3 - x^2y = 2?

Back

Differentiate both sides with respect to x, applying the product rule and implicit differentiation to get dy/dx = 2y(x - y^2)/(x(6y^2 - x)).

Front

What is the derivative of xy = 6?

Back

Using implicit differentiation: dy/dx = -y/x.

Front

What is the derivative of x + 4y = 1?

Back

Using implicit differentiation: dy/dx = -1/4.

Front

What is the product rule in differentiation?

Back

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

Front

What is the chain rule in differentiation?

Back

The chain rule states that if a function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

What is the quotient rule in differentiation?

Back

The quotient rule states that if u and v are functions of x, then d(u/v)/dx = (u'v - uv')/v^2.

Front

What is the significance of implicit differentiation?

Back

Implicit differentiation allows us to find derivatives of functions that are not easily solvable for y, especially in cases where y is defined implicitly.