Guide on Implicit Differentiation
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Motivating example
In Calculus, we are often given a function
This is clearly not of form
Even though
We know from
Taking the derivative of both sides gives:
The right-hand side is easy - the derivative of a constant is
Let's now tackle the left-hand side:
We must use the chain rule when taking the derivative of
We now substitute
Now that we are done with taking the derivative, we can drop the
Finally, let's make
Unlike the standard case, the derivative of
Computing slope of tangent line of a circle using implicit differentiation
Consider a circle with radius
Compute the slope of the tangent line at
Solution. Let's first confirm that the point
Great, so the point
Let's substitute
This is the slope of the tangent line at
Finally, let's illustrate our findings:

Great, we're done!
Performing implicit differentiation
Consider the following function:
Compute the slope of the tangent line at point
Solution. We will tackle this problem in two ways:
standard way of differentiation
implicit differentiation
Unlike the circle example, we can easily make
Taking the derivative of
Therefore, at point
Let's now tackle this problem using implicit differentiation. We take the derivative of both sides with respect to
Therefore, at point
This is the same as the slope computed using the standard way of differentiation!
Finally, let's visualize the tangent line:

Great, we're done!