Part 2/6:
While it is possible to rearrange the first equation to express y as ( y = \pm\sqrt{25 - x^2} ), this explicit formulation is not always feasible, especially with more complex functions. Here is where implicit differentiation comes to our aid.
The Process of Implicit Differentiation
To differentiate an implicit function, we take the derivative of both sides of the equation with respect to x. This is based on the principle that if two quantities are equal, their derivatives are also equal. Therefore, starting with an equation like ( x^2 + y^2 = 25 ), we apply the derivative to both sides:
[
\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)
]
The left-hand side simplifies using the power rule and chain rule to give us:
[
2x + 2y \frac{dy}{dx} = 0
]