Monday, February 24, 2014

Advanced Math Solutions – Derivative Calculator, Implicit Differentiation

We’ve covered methods and rules to differentiate functions of the form y=f(x), where y is explicitly defined as a function of x (click here to review explicit differentiation).  But what if we have to derive functions that are not set this way?  If you can easily express y as a function of x, by all means do that first.  For example x²+y=1, isolate y as a function of x:  y= (1-x²) and use the derivative rules.  Let’s look at x²+y²=1,  or y=sin(3x+4y), clearly isolating y is not trivial, this is where we’ll be using implicit differentiation;  Derive the left hand side and the right hand side with respect to x, and isolate y’.  It is basically an application of the chain rule, just remember that y is not a constant, it is a function of x.


Let’s see how it works (click here):


Now let’s take a closer look how to differentiate sin(3x+4y(x)) with respect to x:


Here’s another example, only this time differentiation with respect to y (click here):


Until next time,
Michal

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