# 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 $f\left(x\right)$ with respect to $x$ is denoted by ${f}^{\mathrm{\prime }}\left(x\right)$ or $\frac{df}{dx}$.

### Basic Differentiation Rules:

1. Power Rule:

• For a function $f\left(x\right)={x}^{n}$, the derivative is ${f}^{\mathrm{\prime }}\left(x\right)=n{x}^{n-1}$.
2. Constant Multiple Rule:

• If $c$ is a constant and $f\left(x\right)$ is differentiable, then $\frac{d}{dx}\left[cf\left(x\right)\right]=c\cdot \frac{d}{dx}\left[f\left(x\right)\right]$.
3. Sum and Difference Rule:

• The derivative of the sum/difference of two functions is the sum/difference of their derivatives: $\frac{d}{dx}\left[f\left(x\right)±g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]±\frac{d}{dx}\left[g\left(x\right)\right]$.
4. Product Rule:

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

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

### Special Derivatives:

• Trigonometric Functions:

• $\frac{d}{dx}\left[\mathrm{sin}\left(x\right)\right]=\mathrm{cos}\left(x\right)$
• $\frac{d}{dx}\left[\mathrm{cos}\left(x\right)\right]=-\mathrm{sin}\left(x\right)$
• $\frac{d}{dx}\left[\mathrm{tan}\left(x\right)\right]={\mathrm{sec}}^{2}\left(x\right)$
• Exponential and Logarithmic Functions:

• $\frac{d}{dx}\left[{e}^{x}\right]={e}^{x}$
• $\frac{d}{dx}\left[\mathrm{ln}\left(x\right)\right]=\frac{1}{x}$

### Chain Rule:

Chain Rule:

• If $y=f\left(u\right)$ and $u=g\left(x\right)$, 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 $f\left(x\right)$ represents the rate of change of its first derivative with respect to $x$. It's denoted as ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)$ 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.