Derivative

Table Of Contents

The derivative is using the Difference Quotient to create a prime function.

Differentiability

For a point to be differentiable and have a derivative, it must:

1. Be continuous at that point
2. $f^{\prime }(x^{-})=f^{\prime }(x)=f^{\prime }(x^{+})$left limit’s slope be the same as the right limit’s slope which is same as the limit’s slope. If there is a sharp jump from one slope to the other, then it is not differentiable

Concepts

• Derivative Formal Definition
• First Principles

Notations

Leibniz Notation

$\frac{dy}{dx}f(x)$ OR $\frac{d^{n}y}{d{x}^n}f(x)$ OR $\frac{d^n}{d{x}^n}f(x)$ OR ${\lim }_{x\to 0}\frac{△y}{△x}f(x)$ OR $\frac{dy}{dx}{|}_{x=a,y=b}$

Prime

$f^{\prime }(x)$

Derivation Rules

These are all theorems.

1. The first part declares the new function is differentiable and that individual derivatives exist
2. The second part give the formula to get the corresponding derivative

List

• Difference Quotient
• Derivative of A Constant
• Derivative Power Rule
• Derivative Constant Multiple
• Derivative Sum Difference
• Derivative Product Rule
  • Derivative Triple Product Rule
  • Derivative Quadruple Product Rule
• Derivative Quotient Rule
• Derivative Chain Rule
• Derivative Power of A Function Rule
• Derivative of Exponential with Euler’s Number Base
• Derivative of General Exponential
• Derivative of Natural Logarithm
• Derivative of General Logarithmic Function
• Derivative of Sine Function
• Derivative of Cosine Function
• Derivative of Tangent Function
• Derivative of Cosecant Function
• Derivative of Secant Function
• Derivative of Cotangent Function
• Trig Inverse Function Derivatives
• Derivative of Inverse Function
• Derivative of Function to a Function Power