Implicit Differentiation Overview

Implicit differentiation is an extension of the same derivative rules and processes you have already learned. The questions you will be asked are still asking you to find the derivative of an equation, the $\frac{\mathit{dy}}{\mathit{dx}}$, or $s\prime \left(t\right)$, $y\prime$, or $f\prime \left(x\right)$ for example. The big difference between these problems and the ones you did previously is that your new problems cannot be made to fit your traditional y = equation with x’s setup. Up to this point the equations you have been working with have always been a nice $y=\sqrt{x^2+3}$, $s\left(t\right)={3t}^2+7t–19$, or $f\left(x\right)=\frac{\mathit{cos}\left(x\right)}{{7x}^2}$. Where the y-piece (dependent variable) is on one side of the equals and all the x’s (independent variables) are on the other side of the equals, or could be solved algebraically to make them look this way.

What we see with implicit differentiation setups are the y’s and x’s are mixed together in the equation, and there is no way of separating them back out to get yourself a nice clean y= equation. Examples might look like: $x^2+y^2=3xy$ or $\frac{p+q}{\sqrt{q}}=5$, where the variables you have in the equation are all mixed together and you can’t separate them from each other.

The results of these types of implicit differentiation problems are the same as previous derivatives. The derivative,no matter how you find it, is a formula for finding the slope of the given equation at a single point, the instantaneous rate of change, the slope of the tangent line. Remember, derivatives are a formula for finding the slope of an equation, even equations that have multiple variables.

Implicit differentiation will require you to play closer attention to some language that you have probably just overlooked previously. With these problems it will be critical to track the language “with respect to”. Up to this point if you were working with an equation with x’s in it, then you would have seen language “with respect to x ”, or if you were working with t’s , then it was “with respect to t ”. This language tells us if we are going to $\frac{d}{\mathit{dx}}$ or $\frac{d}{\mathit{dt}}$ our equation. These “with respect to” variables always matched up with our problems independent variable, so we never worried about it. When we cannot separate out our variables, this language will have an effect on every piece of our equation’s derivative process.

Implicit differentiation will be the technique you need to apply when doing what are called Related Rates word problems. These are basically word problems that want you to find how some portion of the problem is changing, usually with respect to time , $\frac{d}{\mathit{dt}}$.