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# Taking derivatives

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

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The derivative & tangent line equations Interpreting derivative challenge

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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

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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

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

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

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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

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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)

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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)

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

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

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Basic derivative rulesDifferentiating polynomialsTangents of polynomialsDifferentiating polynomials review

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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

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

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

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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

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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

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

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

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Differentiating rational functionsDifferentiating rational functions review

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

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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

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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

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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

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

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

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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

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

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

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Disguised derivatives

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

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Composite exponential function differentiationWorked example: Composite exponential function differentiation

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Vector-valued functions introVector-valued functions differentiationParametric equations differentiation

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Second derivativesSecond derivatives review

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

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