As the Advanced Placement (AP) testing season nears this spring, it is important to familiarize ourselves with the content matter. According to College Board, differentiation: implicit, composite, and inverse functions will be 9-13% of the test score in 2022.
To develop a better understanding of implicit differentiation, we need a brief semantic breakdown of the two types of functions: implicit and explicit. We must recognize that “implicit” and “explicit” are antonyms, while “explicit” means “clearly defined or expressed,” “implicit” means the latter. In the following sections, we will touch upon concepts like functions, derivatives, and the chain rule, all of which are pertinent to the AP Calculus AB Exam.
What do we do when y isn’t clearly expressed as a function of x, or vice versa, much like in implicit functions? Well, there are three main steps to successfully differentiate an equation implicitly. Firstly, we take the derivative. Secondly, we gather all terms with dy/dx onto the left side of the equation. Thirdly, we factor out dy/dx if necessary, to create a single dy/dx term. Fourthly, we solve for dy/dx.
Let’s have a look at a sample problem: “find dy/dx for y² – 5x³ = 3x.”
In step 1, we take the derivative.
2y dy/dx – 15x² = 3 dy/dx
In step 2, we gather all terms with dy/dx onto the left side of the equation.
2y dy/dx – 3 dy/dx = 15x²
In step 3, we factor out dy/dx if necessary, to create a single dy/dx term.
dy/dx (2y – 3) = 15x²
In step 4, we solve for dy/dx.
dy/dx = 15x²/2y – 3
Hopefully, this has facilitated your use of implicit differentiation as a method to differentiate implicitly defined functions, and you are now able to complete your exam with a firm resolve.
By: Estefania Olaiz
March 20, 2022