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How to Differentiate Implicitly

In most differentiation problems, y is written explicitly as a function of x, which just means that the equation is solved for y; in other words, y is by itself on one side of the equation. Sometimes, however, you are asked to differentiate an equation that’s not solved for y, like

image0.png

This equation defines y implicitly as a function of x, and you can’t write it as an explicit function because it can’t be solved for y. For such a problem, you need implicit differentiation. Here’s how it works:

image1.png
  1. Differentiate the terms on both sides of the equation one at a time.

    When you differentiate implicitly, you treat terms that don’t contain any ys in the ordinary way. So for the above equation, you differentiate the second and third terms in the regular way: the power rule for the second term and the trig rule for the third term. The fact that this is an implicit differentiation problem has no effect whatsoever on how you deal with those two terms.

    For terms containing one or more ys, however, you treat the ys like you treat the stuff in a chain rule problem. All you do is differentiate the y-term in the regular way but then multiply that result by y prime. It’s a piece of cake.

    So, for the first and fourth terms in the above equation, you just use the regular power rule, and then tack on a y prime when you’re done. Here’s what you should have so far:

    image2.png
  2. Collect all terms containing a y´ on the left side of the equation and all other terms on the right side.

    image3.png
  3. Factor out y´.

    image4.png
  4. Divide for the final answer.

    image5.png

Note that this derivative is expressed in terms of x and y, instead of just x. So, if you want to evaluate the derivative to get the slope at a particular point, you need to have values for both x and y that you can plug into the derivative.

image6.png

Either way is fine. Take your pick.

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