A Guide to Implicit Differentiation: Techniques for Finding Derivatives of Implicit Functions

Implicit Differentiation

take derivative w/ respect to x, solve for dy/dx

Implicit differentiation is a technique used in calculus to find the derivatives of functions that are not expressed in the explicit form y = f(x), where y is the dependent variable and x is the independent variable.

Instead, implicit functions are often represented by equations containing both dependent and independent variables, such as x^2 + y^2 = 25. With implicit differentiation, we can find the derivative dy/dx, also known as the derivative of y with respect to x, even when y is not explicitly expressed as a function of x.

To perform implicit differentiation, we follow these steps:

1. Differentiate each term of the equation with respect to x as usual, treating y as a dependent variable. For example, differentiating x^2 with respect to x gives 2x, and differentiating y^2 with respect to x gives 2y * dy/dx using the chain rule.

2. Keep all the terms involving dy/dx on one side of the equation and move the other terms to the opposite side. In the given example, we would move 2x to the opposite side of the equation, resulting in 2y * dy/dx = -2x.

3. Solve for dy/dx by dividing the terms involving dy/dx by the other terms. In our example, dividing both sides by 2y gives dy/dx = -x/y.

So, the derivative of the implicit function x^2 + y^2 = 25 with respect to x is dy/dx = -x/y.

Implicit differentiation allows us to find derivatives in scenarios where it might be challenging to solve for y explicitly. It is particularly useful in cases involving curves and functions that cannot be easily expressed as explicit functions.

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