Part 4/6:
Now we have two possible values for y: 4 or -4. This leads to two potential results for ( \frac{dy}{dx} ):
[
\frac{dy}{dx} = -\frac{3}{4} \text{ or } \frac{3}{4}
]
The Utility of Implicit Differentiation
Implicit differentiation shines especially in more complex equations such as ( x^3 + y^3 = 6xy ). The process is similar:
Differentiate both sides with respect to x.
Apply the product rule as necessary and combine like terms.
Upon differentiating, you arrive at an expression that can be solved for ( \frac{dy}{dx} ) without isolating y.