• anonymous
Can somebody explain implicit differentiation and what it can be used for?
MIT 18.02 Multivariable Calculus, Fall 2007
  • Stacey Warren - Expert
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  • katieb
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  • anonymous
short answer: take the equation: y = sin(3x +4y). how can we take the derivative here? the x and y terms are 'mixed in' together. implicit differentiation is a technique to take the derivative here without having to change the equation around first. this may be confusing, so the longer answer below clarifies! first, let's consider ''regular'' differentiation. we have some function y=f(x) where y is alone on the left and there are only x terms on the right. we calculate dy/dx (also known as f'(x)) using all the tools we've learned--product rule, chain rule, etc. notice that y is written alone on one side and the 'function' written out on the other. this is called writing a function "explicitly" because y is shown "explicitly" (clearly) to be equal to the function written out on the other side. now, consider instead a more complicated equation where y and x terms are mixed up and appear on both sides of the equation! for example, the equation of a circle is often written this way: \[x ^{2} + y ^{2} = r ^{2}\]notice that the value of y is not written out explicitly or clearly. to know exactly what y is in terms of x, we would have to solve for y. this would be ONE way to take the derivative of the function: solve for y and then to take the derivative as we normally would. HOWEVER, there are two drawbacks to this method. first - we have to solve for y before we even get started with our real goal, taking the derivative. second - sometimes we want the derivatives of equations which are really hard to solve for y--even impossible! in this case, it would be better to be able to take the derivative without having to solve for y first. AHA! this is what implicit differentiation is: taking the derivative of a or relation when y is not already alone on one side of the equation. for example: y=sin(3x +4y). HOW would you take the derivative? solving for y? NO WAY! we want a technique to take the derivative of this function AS IT IS WRITTEN. that technique is called implicit differentiation.

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