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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

    Logarithmic functions

    Conclude by giving your students these challenges:

    • Penta Colour by NRICH
    • Root to Poly by NRICH
    • Rainstorm Sudoku by NRICH
    • Exact Dilutions by NRICH
    • Pencil Prices

    Find \(x{:}\)

    There are two elegant ways to solve this problem, both of which can be done mentally. Here's the first way: Notice that \(ACEDB\) is a pentagon, and thus, it's interior angles sum to \(540^\circ.\) Notice that \(\angle DEC = 360^\circ - 160^\circ = 200^\circ.\) Adding up the known angles in the pentagon, gives us \(80^\circ + 20^\circ + 200^\circ + 10^\circ = 310^\circ.\) But we need \(540^\circ,\) so the missing angle, \(\angle BDE,\) must be \(230^\circ.\) Thus, \(x = 360^\circ - 230^\circ = 130^\circ.\) Here's the second way: Start by drawing \(\overline{BC}.\) Notice that \(\triangle ABC\) has \(80^\circ + 20^\circ + 10^\circ = 110^\circ,\) so it's lacking \(70^\circ.\) Also, notice that \(BDEC\) is a quadrilateral, and thus, it's interior angles sum to \(360^\circ.\) Adding up the known angles in the quadrilateral, gives us \(160^\circ + 70^\circ = 230^\circ,\) but we need \(360^\circ,\) so \(x = 360^\circ - 230^\circ = 130^\circ.\)

    YouTube videos

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    • 1257Calculus of Logarithmic Functions (4 of 4: The importance of simplifying first)
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    • 4272Logarithmic Differentiation of (sin x)^x ❖ Calculus 1
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