What is implicit differentiation?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Excellent question — and that’s exactly what you’re going to learn in today’s calculus class.
Let’s go!
Did you know that implicit differentiation is just a method for taking the derivative of a function when x and y are intermixed?
Implicit Vs Explicit Functions
But to really understand this concept, we first need to distinguish between explicit functions and implicit functions.
An explicit function is an equation written in terms of the independent variable, whereas an implicit function is written in terms of both dependent and independent variables.
Explicit Vs. Implicit
Notice how all the explicit functions are solved for one variable (i.e., one variable on the left-hand-side and every other term is on the right) while the implicit functions have the variables all intermixed on both sides of the equation. This does not leave an easy way for us to solve for one variable.
How To Do Implicit Differentiation
In all of our previous derivative lessons, we have dealt with explicit functions only, as they are already solved for one variable in terms of another. But now it’s time to learn how to find the derivative, or rate of change, of equations that contain one or more variables and when x and y are intermixed.
- Take the derivative of every variable.
- Whenever you take the derivative of “y” you multiply by dy/dx.
- Solve the resulting equation for dy/dx.
Example
Let’s use this procedure to solve the implicit derivative of the following circle of radius 6 centered at the origin.
Implicit Differentiation Example – Circle
And that’s it!
The trick to using implicit differentiation is remembering that every time you take a derivative of y, you must multiply by dy/dx.
Furthermore, you’ll often find this method is much easier than having to rearrange an equation into explicit form if it’s even possible.
Example
Let’s look a harder problem with trig where x’s and y’s are intermixed.
Implicit Derivative – Trig And Exponential Functions
Example
And sometimes, we will experience implicit functions with more than one y-variable. All this means is that we will have multiple dy/dx terms that we will need to collect in order to simplify, as the following example nicely highlights.
Differentiate Implicitly – Product Rule
Not so bad, right?!
Implicit Differentiation Worksheets (PDF)
Let’s put that pencil to paper and try it on your own.
- Implicit Differentiation Practice: Improve your skills by working 7 additional exercises with answers included.
Video Tutorial w/ Full Lesson & Detailed Examples (Video)
Together, we will walk through countless examples and quickly discover how implicit differentiation is one of the most useful and vital differentiation techniques in all of calculus.
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