Derivatives: definition and basic rules videos

Newton, Leibniz, and Usain Bolt

Derivative as a concept

Derivative as slope of curve

Secant lines & average rate of change

The derivative & tangent line equations

Slope of a line secant to a curve

Secant line with arbitrary point

Secant line with arbitrary difference

Secant line with arbitrary difference (with simplification)

Secant lines: challenging problem 1

Secant line with arbitrary point (with simplification)

Secant lines: challenging problem 2

Formal definition of the derivative as a limit

Worked example: Derivative as a limit

Worked example: Derivative from limit expression

Formal and alternate form of the derivative

The derivative of x² at x=3 using the formal definition

The derivative of x² at any point using the formal definition

Limit expression for the derivative of function (graphical)

Estimating derivatives

Differentiability at a point: algebraic (function is differentiable)

Differentiability at a point: graphical

Differentiability and continuity

Differentiability at a point: algebraic (function isn't differentiable)

Power rule

Power rule (with rewriting the expression)

Basic derivative rules: find the error

Basic derivative rules

Basic derivative rules: table

Differentiating polynomials

Differentiating integer powers (mixed positive and negative)

Tangents of polynomials

Worked example: Derivatives of sin(x) and cos(x)

Derivatives of sin(x) and cos(x)

Derivative of 𝑒ˣ

Derivative of ln(x)

Worked example: Product rule with table

Differentiating products

Product rule

Worked example: Product rule with mixed implicit & explicit

Worked example: Quotient rule with table

Quotient rule

Differentiating rational functions

Normal to y=𝑒ˣ/x²

Tangent to y=𝑒ˣ/(2+x³)

Derivatives of tan(x) and cot(x)

Derivatives of sec(x) and csc(x)

Proof: Differentiability implies continuity

Proof of power rule for positive integer powers

Proof of power rule for square root function

Justifying the power rule

Proof of the derivative of sin(x)

Limit of (1-cos(x))/x as x approaches 0

Limit of sin(x)/x as x approaches 0

Proof of the derivative of cos(x)

Product rule proof