Understanding the Derivatives of Implicit Functions | A Step-by-Step Guide

Derivatives of implicit functions?

Derivatives of implicit functions refer to the process of finding the rate of change of a dependent variable with respect to an independent variable when the relationship between the variables is not explicitly defined

Derivatives of implicit functions refer to the process of finding the rate of change of a dependent variable with respect to an independent variable when the relationship between the variables is not explicitly defined. In other words, it involves finding the derivative of an equation that does not explicitly solve for the dependent variable.

To find the derivative of an implicit function, you can use the implicit differentiation technique. Here’s a step-by-step process:

1. Write down the equation representing the implicit function. For example, let’s consider the equation: x^2 + y^2 = 25.

2. Differentiate both sides of the equation with respect to the independent variable (usually denoted as x or t). For example, differentiating x^2 + y^2 = 25 with respect to x gives: 2x + 2yy’ = 0.

3. Isolate the derivative term (y’) on one side of the equation. In our example, isolate y’: 2yy’ = -2x.

4. Divide both sides of the equation by 2y to solve for y’: y’ = -x/y.

This derived equation, y’ = -x/y, represents the derivative of the implicit function.

It is important to note that when differentiating an implicit function, the derivative of the dependent variable (y’) is expressed in terms of both the independent variable (x) and the dependent variable itself (y). This is due to the fact that the equation does not explicitly solve for y in terms of x.

Implicit differentiation is commonly used in cases where it is challenging or not feasible to solve the equation explicitly for y. It allows us to find the derivative of the dependent variable even in complex equations or systems of equations.

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