#Derivatives of Implicit Functions

#Table of Contents

1. Introduction to Implicit Functions
2. Implicit Differentiation
3. Examples and Applications
4. Differentiating Inverse Functions
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

#Introduction to Implicit Functions

#What is an Implicit Function?

• An equation in the form $y=f(x)$ or $x=f(y)$ is written explicitly.

• Equations involving both $x$ and $y$ are referred to as implicit functions.

• For such equations, we cannot express $y$ solely in terms of $x$ or vice versa.
• However, these equations define a relationship between $x$ and $y$.

#Implicit Differentiation

#What is Implicit Differentiation?

• Implicit differentiation is the method used to differentiate implicit functions.
• To differentiate an implicit function with respect to $x$, each term is differentiated with respect to $x$.
• For terms involving only $x$, this is straightforward.
• For terms involving $y$, we apply the chain rule.

• Once each term has been differentiated, rearrange the equation to solve for $\frac{dy}{dx}$.
• Factorize out $\frac{dy}{dx}$ if necessary.
• Substitute specific $(x, y)$ values to find the derivative at a point if required.

#How to Use Implicit Differentiation?

#Example

Differentiate $x^2 + y^2 =...