Part 2/5:
It’s important to be cognizant of the domain of the inverse cosine function, which is defined for values typically between ( 0 ) and ( \pi ). Understanding the behavior of these functions within this domain is crucial for accurately calculating derivatives.
Deriving the Derivative Using Implicit Differentiation
To find the derivative of ( y ) with respect to ( x ), we apply implicit differentiation to the equation ( \cos(y) = x ).
Taking the derivative of both sides, we obtain:
[
-\sin(y) \frac{dy}{dx} = 1
]
Rearranging this equation yields:
[
\frac{dy}{dx} = -\frac{1}{\sin(y)}
]