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    • ▾Calculus 1
      • ▸Limits and continuity
        • •Intro to calc 1
        • •Estimating limits from graphs
        • •Estimating limits from tables
        • •Intro to continuity
        • •Finding and classifying discontinuities
        • •Finding the value that makes the function continuous
      • ▾Derivatives: definition and basic derivative rules
        • •Proving differentiable implies continuous
        • •Derivative of a function from first principles
        • •Finding the tangent or normal line through the given point
        • •Constant rule
        • •Constant multiple rule for derivatives
        • •Sum, difference, and power rules for derivatives
        • •Product rule
        • •Proving the product rule
        • •Quotient rule
        • •Derivatives of trig functions
        • •Proving properties of even and odd functions (calculus 1)
        • •Leibniz's derivative notation
        • •Logarithmic functions
      • ▸Derivatives: composite, implicit, and inverse functions
        • •Derivative at a point from first principles
        • •Derivatives of exponential functions
        • •Derivatives of logarithmic functions
        • ▸Derivatives of hyperbolic and inverse hyperbolic functions
          • •Using the derivatives of the hyperbolic functions
          • •Proving the derivatives of the hyperbolic functions
          • •Proving the derivatives of the inverse hyperbolic functions
        • •Derivatives of inverse trig functions
        • •Derivatives of transcendental functions
        • •Derivatives of trigonometric functions under transformations
        • •Derivative using u-substitution
        • •Determining derivatives from graphs
        • •Higher order derivatives
        • •Chain rule
        • •Derivatives of inverse functions
        • •Choosing the derivative rules to use
      • ▸Applying derivatives to analyze functions
        • •Applications of trigonometric derivatives
        • •Extreme value theorem
        • •First derivative test
        • •Mean value theorem
        • ▸Monotonic intervals and functions
          • •Intro
          • •Proofs
        • •Related rates
        • •Second derivative test
        • •Concavity
        • •Factoring a cubic polynomial with a double root
        • •Graphing using derivatives
        • ▸Special points
          • ▸Finding critical values from graphs
            • •Lesson
            • •Practice
          • •Finding critical values using derivatives
          • ▸Inflection points
            • •Graphically finding points of inflection
            • •Using the second derivative to find points of inflection
          • •Classifying stationary points
          • •Sketching polynomials using stationary points
        • •Bisection method
        • •Newton's method
        • •False position method
        • •L'Hopital's rule
      • ▸Integrals
        • •Basic integration problems
        • ▸Integration by u-substitution
          • ▸Integration by u-substitution (less difficult)
            • •Lesson
            • •Practice
          • •Integration by u-substitution (more difficult)
          • •Integrating exponential functions by u-substitution
        • •Visually determining antiderivative
      • ▸Applications of integrals
        • •Proving the formula for the area of a circle
        • •Area between two curves
        • ▸Solids of revolution
          • •Disc and washer methods (circular cross sections)
          • •Volumes of solids with known cross sections
          • •Shell method
          • •Volume of a sphere (calculus \(2\))
          • •Volume of a cone (calculus 2)
     › Calculus 1 › Derivatives: definition and basic derivative rules › Quotient rule

    Using the theorem

    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{6x^2}{\ln x}\right)\)
    \(\dfrac{6x(2\ln x - 1)}{(\ln x)^2}\)
    70576Calculus Derivative 6x^2 / ln x Quotient Rule (Test Question)
    Prof. Redden
    Find the derivative:  \(\dfrac{d}{dx}\left[x^5(1 - x^6)\right]\)
    \(5x^4 - 11x^10\)
    860Quotient Rule (2 of 2: Worked examples)
    Eddie Woo
    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{5 - 2x}{x^2 + 1}\right)\)
    \(\dfrac{2x^2 - 10x - 2}{\left(x^2 + 1\right)^2}\)
    860Quotient Rule (2 of 2: Worked examples)
    Eddie Woo
    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{x^3 - 4x}{2x^2 + 1}\right)\)
    \(\dfrac{\left(3x^2 - 4\right)\left(2x^2 + 1\right) - 4x\left(x^3 - 4x\right)}{\left(2x^2 + 1\right)^2}\)
    27132.7 Quotient Rule 01
    rootmath
    27142.7 Quotient Rule 02
    rootmath
    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{e^{2x} - 1}{x^3}\right)\)
    \(\dfrac{x^3\left(2e^{2x}\right) - 3x^2\left(e^{2x} - 1\right)}{\left(x^3\right)^2}\)
    27132.7 Quotient Rule 01
    rootmath
    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{4x - 2}{x^2 - 1}\right)\)
    \(\dfrac{-4x^2 + 4x - 4}{\left(x^2 - 1\right)^2}\)
    13741How to take the derivative using the quotient rule and simplifying the numerator
    Brian McLogan
    Find the derivative:  \(\dfrac{d}{dx}\left[\left(\dfrac{x + 4}{x + 3}\right)(2x + 5)\right]\)
    \(\dfrac{2x^2 + 16x + 29}{(x + 3)^2}\)
    13657Find the derivative using quotient rule inside of the product rule
    Brian McLogan
    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{\cos x}{x}\right)\)
    \(\dfrac{-x\sin x - \cos x}{x^2}\)
    13156Learn how use the quotient rule to take the derivative including cosine
    Brian McLogan
    Find the derivative:  \(\dfrac{d}{dx}\left(\dfrac{3 - \dfrac{1}{x}}{x + 5}\right)\)
    \(\dfrac{-3x^2 + 2x + 5}{\left(x^2 + 5x\right)^2}\)
    20437Use the quotient rule to take the derivative
    Brian McLogan
    Find the derivative:  \(\dfrac{d}{dx}\left[x\left(1 - \dfrac{4}{x + 3}\right)\right]\)
    \(1 - \dfrac{12}{(x + 3)^2}\)
    16444Learn how to take derivative using quotient rule by simplifying first
    Brian McLogan