Implicit Differentiation

Front

What is Implicit Differentiation?

Back

A technique used to find the derivative of a function defined implicitly by an equation involving both x and y.

Front

What is the derivative of x^3 + y^3 = 36?

Back

dy/dx = -x^2/y^2.

Front

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

Back

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

Front

What is the derivative of sin(y^2) = x?

Back

dy/dx = 1/(2y cos(y^2)).

Front

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

Back

dy/dx = -y/(x + 2y).

Front

What is the Chain Rule in differentiation?

Back

A rule for finding the derivative of a composite function: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

What is the product rule in differentiation?

Back

A rule for differentiating products of functions: if u and v are functions of x, then d(uv)/dx = u'v + uv'.

Front

What is the quotient rule in differentiation?

Back

A rule for differentiating quotients of functions: 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

It allows us to differentiate equations where y cannot be easily isolated.

Front

How do you apply implicit differentiation to find dy/dx?

Back

Differentiate both sides of the equation with respect to x, treating y as a function of x, and solve for dy/dx.