# 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 $F\left(x,y\right)=0$ or $G\left(x,y\right)=0$, 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:

1. Differentiate both sides of the equation with respect to $x$:
• Apply the chain rule whenever $y$ appears.
2. Solve for $\frac{dy}{dx}$:
• Collect terms involving $\frac{dy}{dx}$ on one side of the equation.

### Example:

Given the equation ${x}^{2}+{y}^{2}=25$:

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

2. 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.