# Taking derivatives

## Introduction to differential calculus

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Newton, Leibniz, and Usain BoltDerivative as slope of curveDerivative & the direction of a functionDerivative notation review

## Derivative as slope of tangent line

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The derivative & tangent line equations Interpreting derivative challenge

## Derivative as instantaneous rate of change

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Tangent slope as instantaneous rate of changeEstimating derivatives with two consecutive secant linesApproximating instantaneous rate of change with average rate of change

## Secant lines

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Slope of a line secant to a curveSecant line with arbitrary differenceSecant line with arbitrary pointSecant line with arbitrary difference (with simplification)Secant line with arbitrary point (with simplification)Secant lines: challenging problem 1Secant lines: challenging problem 2

## Derivative as a limit

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Formal definition of the derivative as a limitDerivative as a limit: numerical

## Formal definition of derivative

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Formal and alternate form of the derivativeWorked example: Derivative as a limitWorked example: Derivative from limit expression
Practice

## Using the formal definition of derivative

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The derivative of x² at x=3 using the formal definitionThe derivative of x² at any point using the formal definitionLimit expression for the derivative of a linear functionLimit expression for the derivative of cos(x) at a minimum pointLimit expression for the derivative of function (graphical)Tangent lines and rates of change

## Differentiability

Learn
Differentiability at a point: graphicalDifferentiability at a point: algebraic (function is differentiable)Differentiability at a point: algebraic (function isn't differentiable)Differentiability at a point (old)

## Derivative as a function

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The graphical relationship between a function & its derivative (part 1)The graphical relationship between a function & its derivative (part 2)Matching functions & their derivatives graphically Connecting f and f' graphicallyMatching functions & their derivatives graphically (old)
Practice

## Review: Derivative basics

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Practice

## Basic differentiation rules

Learn
Basic derivative rules (Part 1)Basic derivative rules (Part 2)Basic derivative rules: find the errorBasic derivative rules: tableBasic differentiation reviewDerivative notation review

## Power rule

Learn
Power ruleJustifying the power ruleProof of power rule for positive integer powersProof of power rule for square root function

## Polynomial functions differentiation

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Basic derivative rulesDifferentiating polynomialsTangents of polynomialsDifferentiating polynomials review

## Rational functions differentiation (intro)

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Differentiating integer powers (mixed positive and negative)Worked example: Tangent to the graph of 1/x

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Fractional powers differentiationRadical functions differentiation introPower rule review

## Sine & cosine derivatives

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Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)Worked example: Derivatives of sin(x) and cos(x)

## eˣ and ln(x) derivatives

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Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
Practice

## Review: Basic differentiation

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## Product rule

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Product ruleDifferentiating productsWorked example: Product rule with tableWorked example: Product rule with mixed implicit & explicitProduct rule to find derivative of product of three functionsProduct rule proofProduct rule review

## Chain rule

Learn
Chain ruleWorked example: Derivative of cos³(x) using the chain ruleWorked example: Derivative of ln(√x) using the chain ruleWorked example: Derivative of √(3x²-x) using the chain ruleChain rule overviewWorked example: Chain rule with tableQuotient rule from product & chain rulesChain rule with the power ruleApplying the chain rule graphically 1 (old)Applying the chain rule graphically 2 (old)Applying the chain rule graphically 3 (old)Chain rule
Practice
4 questions

## Chain rule proof

Learn
Proof: Differentiability implies continuityIf function u is continuous at x, then Δu→0 as Δx→0 Chain rule proof

## Quotient rule

Learn
Quotient rule from product & chain rulesWorked example: Quotient rule with tableTangent to y=𝑒ˣ/(2+x³)Normal to y=𝑒ˣ/x²Quotient rule review

## Review: Product, quotient, & chain rule

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## Rational functions differentiation

Learn
Differentiating rational functionsDifferentiating rational functions review

Learn
Worked example: Derivative of ∜(x³+4x²+7) using the chain rule

## Trigonometric functions differentiation

Learn
Derivatives of tan(x) and cot(x)Derivatives of sec(x) and csc(x)Worked example: Derivative of sec(3π/2-x) using the chain ruleDerivative of tan(x) (old)Differentiating trigonometric functions review

## Exponential functions differentiation

Learn
Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)Derivative of aˣ (for any positive base a)Worked example: Derivative of 7^(x²-x) using the chain ruleDerivative of 2ˣ (old)Differentiating exponential functions review

## Logarithmic functions differentiation

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Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)Derivative of logₐx (for any positive base a≠1)Worked example: Derivative of log₄(x²+x) using the chain ruleDifferentiating logarithmic functions using log propertiesDerivative of logarithm for any base (old)Differentiating logarithmic functions review

## Derivatives capstone

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Derivative of eᶜᵒˢˣ⋅cos(eˣ)Derivative of sin(ln(x²))Common derivatives reviewDerivative rules review

## Implicit differentiation introduction

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Implicit differentiationWorked example: Implicit differentiationWorked example: Evaluating derivative with implicit differentiationShowing explicit and implicit differentiation give same result
Practice

Learn
Implicit differentiation (advanced example)Implicit differentiation (advanced example)Implicit differentiation (advanced example)Derivative of ln(x) from derivative of 𝑒ˣ and implicit differentiationImplicit differentiation review

## Inverse trig functions differentiation

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Derivative of inverse sineDerivative of inverse cosineDerivative of inverse tangentDifferentiating inverse trig functions review

## Derivatives of inverse functions

Learn
Derivatives of inverse functions: from equationDerivatives of inverse functions: from table

## Disguised derivatives

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Disguised derivatives
Practice

## Proofs for the derivatives of eˣ and ln(x)

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Proof: The derivative of 𝑒ˣ is 𝑒ˣProof: the derivative of ln(x) is 1/x

## Logarithmic differentiation

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Composite exponential function differentiationWorked example: Composite exponential function differentiation
Practice

## Parametric & vector-valued function differentiation

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Vector-valued functions introVector-valued functions differentiationParametric equations differentiation

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## Higher-order derivatives

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Second derivativesSecond derivatives review
Practice

## Higher-order derivatives (parametric & vector-valued functions)

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Second derivatives (vector-valued functions)Second derivatives (parametric functions)

## Polar curve differentiation

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Practice