3-2 Implicit Differentiation

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x, rather than explicitly as y = f(x).

Front

How do you find dy/dx using implicit differentiation?

Back

To find dy/dx using implicit differentiation, differentiate both sides of the equation with respect to x, treating y as a function of x, and then solve for dy/dx.

Front

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

Back

The derivative of y^2 with respect to x is 2y(dy/dx) due to the chain rule.

Front

What is the chain rule in differentiation?

Back

The chain rule states that if a function y is composed of another function u, then the derivative of y with respect to x is dy/dx = dy/du * du/dx.

Front

How do you differentiate the equation xy + y^2 = 2?

Back

Differentiate both sides: d/dx(xy) + d/dx(y^2) = d/dx(2). This gives y + x(dy/dx) + 2y(dy/dx) = 0.

Front

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

Back

Differentiating gives 3x^2 + 3y^2(dy/dx) = 0, which can be rearranged to find dy/dx.

Front

What does dy/dx represent in a function?

Back

dy/dx represents the rate of change of y with respect to x, or the slope of the tangent line to the curve at a given point.

Front

How do you find dy/dx at a specific point using implicit differentiation?

Back

Substitute the coordinates of the specific point into the differentiated equation and solve for dy/dx.

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 at that point.

Front

What is the formula for the derivative of a product of two functions?

Back

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