Implicit differentiation is a powerful technique used to find the derivative of functions that are not explicitly defined in terms of one variable, given that a variable can be written as a function of the other. This method is particularly useful when dealing with equations where it's difficult or impossible to isolate one variable.
Assume $y$ is a function of $x$. Sometimes the explicit relationship $y=f(x)$ isn't given, yet we can follow the steps below to differentiate implicitly.
Assume you are interested in finding the derivative with respect to $x$, that is, $\frac{dy}{dx}$.
Let's consider the equation of a circle: $x^2 + y^2 = r^2$
To find $\frac{dy}{dx}$:
The term $\frac{dx}{dx}$ disappears above since it is just equal to 1.
Hence make sure to derive $x$ and $y$ normally and apply the appropriate chain rule to each depending on what you are differentiating with respect to.
It can also be used when they aren't functions of each other,
Related rates problems involve finding the rate of change of one quantity with respect to time when it's related to another quantity whose rate of change is known. These problems often require the use of implicit differentiation and the chain rule.
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