This lesson will take you through the method of implicit differentiation with multiple examples and a quiz at the end to test your knowledge. Implicit differentiation utilizes all of your basic derivative rules to find the derivative of equations that are not in standard form.
What Is Implicit Differentiation?
Up to this point in calculus, most functions that have been derived were in explicit form.Explicit form is the standard y = 2x + 5 or any other function where y is on one side of the equal sign and x is on the other!Implicit Functions are different in that x and y can be on the same side.A simple example is: xy = 1. It is here that implicit differentiation is used.Remember! You HAVE used all of these derivative rules before! y has just been isolated for you.
What happens when you take the derivative of y = 2x? Let’s take a look:
We also know that y‘ is just another way to write dy/dx which means derivative with respect to x.Every time, we take the derivative of a y variable, we will write either y‘ or dy/dx.
REMEMBER: All other derivative rules still apply.
Ex 1: Find dy/dx given that y^3 + 2y^2 – 3y +x^2 = -2 (Written out in the image below)
To begin, let’s take the derivative of y^3. The 3 moves down in front of the y and the exponent decreases by 1… just like our standard derivative!
That’s great! Now we can use the same method to derive the rest of the y variables.
Notice how every term that had a y now has a dy/dxTo finish taking the derivative, we have to find dy/dx of x^2 and of -2. Here, everything is normal.dy/dx of x^2 = 2x and dy/dx of -2 = 0. Putting all of our individual derivatives together, we get:
But are we done? The answer to that is NO. To finish properly taking a derivative implicitly, we need to solve for dy/dx.
To do this, treat dy/dx like it is just another variable and use your algebra skills to isolate it and solve for it.First: Subtract anything that does not have dy/dx to the other side of the equal sign.Second: Factor out dy/dx.Third: Divide both sides by what remains after you factored.
Derive -; Isolate -; SolveEx: 2 Find dy/dx given xy – y = 3Any time you have two variables touching, you need to use the Product Rule
Implicit differentiation is as simple as ‘normal’ differentiation.
In fact, all you have to do is take the derivative of each and every term of an equation. You just have to remember that the derivative is with respect to x. This means that there is an extra step when deriving a term with a y variable.Here are some simple guidelines to follow.1. Differentiate both sides of an equation with respect to x2. Whenever 2 variables are being multiplied or divided, use the respective derivative rule.
(EX: xy, x/y, 4xy^2, etc. all involve either a product or quotient rule.)3. Collect all terms that involve dy/dx on one side of the equation and all other terms on the other side.4. Treat dy/dx like a variable.
Factor dy/dx out of the equation.5. Solve for dy/dx.Now you can differentiate implicitly!
- Explicit form – the standard y = 2x + 5 or any other function where y is on one side of the equal sign and x is on the other
- Implicit functions – are different in that x and y can be on the same side
Upon completion of the lesson on implicit differentiation, confirm your ability to:
- Understand implicit differentiation
- Contrast explicit form and implicit function
- Give examples of implicit differentiation