Question

What is Implicit differentiation?

Answer 1

Implicit differentiation is a technique used in calculus to differentiate equations where the dependent variable and the independent variable are not explicitly separated. In other words, the dependent variable is implicitly defined as a function of the independent variable. This approach is particularly useful when it is difficult or inconvenient to isolate one variable and obtain an explicit function.

To apply implicit differentiation, you differentiate both sides of the equation with respect to the independent variable, usually denoted as \(x\), while treating the dependent variable, often denoted as \(y\), as an implicit function of \(x\). When differentiating any term involving \(y\), you apply the chain rule by multiplying the term by \(\frac{d y}{d x}\) (or \(y^{\prime}\), depending on notation).

Example:
Consider the equation of a circle with radius \(r\) and center at the origin:
\[
x^2+y^2=r^2
\]
Here, it's not easy to isolate \(y\) and express it as an explicit function of \(x\). Instead, we apply implicit differentiation with respect to \(x\) :
\[
\frac{d}{d x}\left(x^2\right)+\frac{d}{d x}\left(y^2\right)=\frac{d}{d x}\left(r^2\right)
\]

Now, differentiate each term:
\[
2 x+2 y \frac{d y}{d x}=0
\]
Finally, solve for \(\frac{d y}{d x}\) :
\[
\frac{d y}{d x}=-\frac{2 x}{2 y}=-\frac{x}{y}
\]
In this example, we've found the derivative \(\frac{d y}{d x}\) without explicitly solving for \(y\) as a function of \(x\). Implicit differentiation is a powerful technique that allows you to find derivatives for equations that are difficult or impossible to solve explicitly for the dependent variable.