Final answer:
Implicit differentiation is a technique used to find the derivative of an equation that cannot be easily solved for y. To perform implicit differentiation, you follow the differentiation rules and treat y as a function of x. Then, you differentiate both sides of the equation with respect to x.
Step-by-step explanation:
Implicit differentiation is a technique used to find the derivative of an equation that cannot be easily solved for y. To perform implicit differentiation, you follow the differentiation rules and treat y as a function of x. Then, you differentiate both sides of the equation with respect to x. This allows you to find the derivative of y with respect to x without explicitly solving for y. Let's walk through an example to illustrate the steps:
Start with a given equation that contains both x and y, such as:
x^2 + 2y = 5
Differentiate both sides of the equation with respect to x:
d/dx(x^2 + 2y) = d/dx(5)
2x + 2(dy/dx) = 0
Isolate dy/dx by rearranging the equation:
2(dy/dx) = -2x
dy/dx = -x
So, the derivative of y with respect to x is -x. This represents the rate of change of y with respect to x. Remember that when you perform implicit differentiation, you are finding the derivative without solving for y explicitly. This can be useful in cases where solving for y is difficult or impossible.