Derivative of Implicit functions
Implicit Functions:
 Implicit functions are equations where the relationship between variables is not explicitly stated. Instead, it's presented as an equation involving both variables.
 Typically represented as $F(x,y)=0$ or $G(x,y)=0$, where $F$ or $G$ could be any expression involving both $x$ and $y$.
Derivative of Implicit Functions:
 Finding the derivative of an implicitly defined function involves using implicit differentiation, treating $y$ as a function of $x$ even when $y$ isn't explicitly written as a function of $x$.
Steps to Find $\frac{dy}{dx}$ Implicitly:
 Differentiate both sides of the equation with respect to $x$:
 Apply the chain rule whenever $y$ appears.
 Solve for $\frac{dy}{dx}$:
 Collect terms involving $\frac{dy}{dx}$ on one side of the equation.
Example:
Given the equation ${x}^{2}+{y}^{2}=25$:

Differentiate both sides with respect to $x$: $\frac{d}{dx}({x}^{2}+{y}^{2})=\frac{d}{dx}(25)$ $2x+2y\cdot \frac{dy}{dx}=0$

Solve for $\frac{dy}{dx}$: $\frac{dy}{dx}=\frac{x}{y}$
Tips:
 Pay attention to chain rule usage:
 Every term involving $y$must be multiplied by $\frac{dy}{dx}$ when differentiating with respect to $x$.
 Isolate the derivative:
 After implicit differentiation, isolate $\frac{dy}{dx}$ if possible to express it explicitly.
 Watch for implicit relationships:
 Implicit derivatives might reveal relationships between $x$ and $y$ not easily seen in explicit functions.
Special Cases:

Higher Derivatives:
 After finding the first derivative $\frac{dy}{dx}$, further derivatives can be found by differentiating the equation again and solving for $\frac{{d}^{2}y}{d{x}^{2}}$ or higher derivatives if needed.

Parametric Forms:
 Sometimes it's helpful to express an implicitly defined function in parametric form, where $x$ and $y$ are defined separately as functions of a parameter.