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 find dy/dx for the equation -7x + 9y = 12?

Back

To find dy/dx, differentiate both sides with respect to x: -7 + 9(dy/dx) = 0. Thus, dy/dx = \frac{7}{9}.

Front

What is the derivative of y with respect to x in the equation x + 4y = 1?

Back

Using implicit differentiation, we get 1 + 4(dy/dx) = 0, leading to dy/dx = -\frac{1}{4}.

Front

Find dy/dx for the equation -7x^3 + 4y^3 = 10.

Back

Differentiating gives -21x^2 + 12y^2(dy/dx) = 0. Thus, dy/dx = \frac{7x^2}{4y^2}.

Front

What is the Product Rule in differentiation?

Back

The Product Rule states that \frac{d}{dx}(fg) = f'g + fg', where f and g are functions of x.

Front

Define a derivative in calculus.

Back

A derivative represents the slope of the tangent line to the curve of a function at a given point.

Front

What is the Chain Rule in differentiation?

Back

The Chain Rule states that if a function y = f(g(x)), then \frac{dy}{dx} = f'(g(x)) \cdot g'(x).

Front

How do you differentiate a constant?

Back

The derivative of a constant is zero.

Front

What is the relationship between implicit differentiation and explicit differentiation?

Back

Implicit differentiation allows us to differentiate equations where y is not isolated, while explicit differentiation applies when y is expressed as a function of x.

Front

What is the significance of the slope of the tangent line?

Back

The slope of the tangent line at a point on a curve represents the instantaneous rate of change of the function at that point.