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Most of our math work thus far has always allowed us to solve an equation for y in terms of x. When an equation can be solved for y we call it an explicit function. But not all equations can be solved for y. An example is:
This equation cannot be solved for y. When an equation cannot be solved for y, we call it an implicit function. The good news is that we can still differentiate such a function. The technique is called implicit differentiation.
. This notation
tells us that we are differentiating with respect to x. Because
y is not native to what are differentiating with respect to, we need to
regard it as a composite function. As you know, when we differentiate a
composite function we must use the chain rule.
Let’s now try to differentiate
the implicit function, 
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This is a "folium of Descartes" curve. This would be very difficulty to solve for y, so we will want to use implicit differentiation. |
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Here we show with Leibnitz notation that we are implicitly differentiating both sides of the equation. |
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On
the left side we need to individually take the derivative of each term.
On the right side we will have to use the product rule. (  ) |
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Here
we take the individual derivatives. Note: Where did the y’ come
from? Because we are differentiating with respect to x, we need
to use the chain rule on the y. Notice that we did use the product
rule on the right side. |
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Now we get the y’ terms on the same side of the equation. |
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Now
we factor y’ out of the expression on the left side. |
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Now
we divide both sides by the  factor
and simplify. |
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We can see in a plot of the implicit function that the slope of the tangent line at the point (3,3) does appear to be -1. |
Another example: Differentiate: 
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Given implicit function |
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Doing implicit differentiation on the function. Note the use of the product rule on the second term |
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We do the algebra to solve for y'. |
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Here we see a portion of plot of the implicit equation with c set equal to 5.. When does it appear that the slope of the tangent line will be zero? It appears to be at about (2.2,2.2). |
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We take our derivative, set it equal to zero, and solve. |
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Now putting x = y in the original implicit equation, we find that... |
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x = y = 2.116343299
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We still must use a computer algebra system to solve this cubic equation. The one real answer is shown at the left. This answer does seem consistent with our visual estimate. This can be done in Maple with the following command: >evalf(solve(x^3-x^2-5=0,x)); |
Links to other explanations of Implicit
Differentiation:
Implicit Differentiation
University of British Columbia
University of Kentucky's Visual Calculus
Implicit Differentiation Using Maple
University of Califonia - Davis
Derivatives of Inverse Trigonometric Functions
Thanks to implicit differentiation, we can develop important derivatives that we could not have developed otherwise. The inverse trigonometric functions fall under this category. We will develop and remember the derivatives of the inverse sine and inverse tangent.
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Inverse sine function. |
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This is what inverse sine means. |
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We implicitly differentiate both sides of the equation with respect to x. Because we are differentiating with respect to x, we need to use the chain rule on the left side. |
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We solve the equation for  . |
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This is because of the trigonometric identity,  . |
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Refer back to the equation in step two above. We have our derivative. |
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The inverse tangent function. |
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This is what inverse tangent means. |
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We implicitly differentiate both sides of the equation with respect to x. Because we are differentiating with respect to x, we need to use the chain rule on the left side. |
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We solve the equation for . |
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This is because of the trigonometric identity,  . |
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Refer back to the equation in step two above. We have our derivative. |
