Differentiation
Introduction to Differentiation:
Definition: Differentiation is a fundamental concept in calculus that involves finding the rate at which a quantity changes concerning another variable. It determines how a function changes as its input changes.
Derivative Notation: The derivative of a function $f(x)$ with respect to $x$ is denoted by ${f}^{\mathrm{\prime}}(x)$ or $\frac{df}{dx}$.
Basic Differentiation Rules:

Power Rule:
 For a function $f(x)={x}^{n}$, the derivative is ${f}^{\mathrm{\prime}}(x)=n{x}^{n1}$.

Constant Multiple Rule:
 If $c$ is a constant and $f(x)$ is differentiable, then $\frac{d}{dx}[cf(x)]=c\cdot \frac{d}{dx}[f(x)]$.

Sum and Difference Rule:
 The derivative of the sum/difference of two functions is the sum/difference of their derivatives: $\frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm \frac{d}{dx}[g(x)]$.

Product Rule:
 For two functions $u(x)$ and $v(x)$, the derivative of their product is: $\frac{d}{dx}[u(x)\cdot v(x)]={u}^{\mathrm{\prime}}(x)\cdot v(x)+u(x)\cdot {v}^{\mathrm{\prime}}(x)$.

Quotient Rule:
 For $u(x)$ and $v(x)$ where $v(x)\mathrm{\ne}0$, the derivative of their quotient is: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{v(x)\cdot {u}^{\mathrm{\prime}}(x)u(x)\cdot {v}^{\mathrm{\prime}}(x)}{[v(x){]}^{2}}$.
Special Derivatives:

Trigonometric Functions:
 $\frac{d}{dx}[\mathrm{sin}(x)]=\mathrm{cos}(x)$
 $\frac{d}{dx}[\mathrm{cos}(x)]=\mathrm{sin}(x)$
 $\frac{d}{dx}[\mathrm{tan}(x)]={\mathrm{sec}}^{2}(x)$

Exponential and Logarithmic Functions:
 $\frac{d}{dx}[{e}^{x}]={e}^{x}$
 $\frac{d}{dx}[\mathrm{ln}(x)]=\frac{1}{x}$
Chain Rule:
Chain Rule:
 If $y=f(u)$ and $u=g(x)$, then the chain rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$.
Implicit Differentiation:
Implicit Differentiation:
 When an equation is given implicitly in terms of $x$ and $y$, you can differentiate both sides of the equation with respect to $x$ and solve for $\frac{dy}{dx}$.
Higher Order Derivatives:
Higher Order Derivatives:
 The second derivative of a function $f(x)$ represents the rate of change of its first derivative with respect to $x$. It's denoted as ${f}^{\mathrm{\prime}\mathrm{\prime}}(x)$ or $\frac{{d}^{2}f}{d{x}^{2}}$.
Applications of Differentiation:
 Optimization: Finding maximum or minimum values of functions.
 Related Rates: Analyzing the rates at which quantities change concerning each other.
 Curve Sketching: Determining behavior and characteristics of a function using derivatives.