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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 › Product rule

    Using the theorem

    Find the derivative of each:

    \(f(x) = 2x\cos x - 2\sin x\)
    19172How to take the derivative using the product rule with sine
    Brian McLogan
    22270Take the derivative of a trigonometric function
    Brian McLogan
    \(h(x) = x^2\sin x\)
    \(h'(x) = 2x \cdot \sin x + x^2 \cdot \cos x\)
    44214How to use the Product Rule for Derivatives Short Video
    The Math Sorcerer
    \(h(x) = x^4e^x\)
    \(h'(x) = 4x^3 \cdot e^x + x^4 \cdot e^x\)
    45694How to Use the Product Rule for Derivatives in Calculus Super Short Video f(x) = x^4*e^x
    The Math Sorcerer
    \(f(x) = 5x^2\sin x\)
    $$\begin{align} f'(x) &= 5x^2 \cdot \cos x + \sin x \cdot 10x \\ &= 5x(x\cos x + 2\sin x) \end{align}$$ Prof. Redden70577
    \(\dfrac{dy}{dx}\left[f^2(0)\right]\) $$\begin{array}{c|c|c} x & f(x) & f'(x) \\ \hline 0 & 9 & -2 \\ 1 & -3 & 1 / 5 \end{array}$$
    $$\begin{align} & \dfrac{dy}{dx}\left[f^2(0)\right] \\[1em] ={} & \dfrac{dy}{dx}\left[f(0)f(0)\right] \\[1em] ={} & f(0)f'(0) + f'(0)f(0) \\[1em] ={} & 2f(0)f'(0) \\[1em] ={} & 2(9)(-2) \\[1em] ={} & -36 \end{align}$$
    21111How to take the derivative using charts
    Brian McLogan
    \(f(x) = x^3\cos x\)
    Brian McLogan15832
    \(h(x) = 2x\sin x\)
    Brian McLogan15878
    \(g(x) = x(5x - 3x^2)\)
    Brian McLogan22654
    \(f(x) = e^{3x}\ln\left(x^2 - 5\right)\)
    Math Meeting4013
    \(f(x) = x^2\sin x\)
    The Organic Chemistry Tutor11511
    \(f(x) = (5x - 9x^3)(8 + x^2)\)
    The Organic Chemistry Tutor11511
    \(f(x) = 4\sin x\tan x\)
    The Organic Chemistry Tutor11511
    \(f(x) = 5x\sin x - x^3\tan x\)
    The Organic Chemistry Tutor11511
    \(R(x) = (x^2 + 6)(7 - 8x)(3x - 5x^3)\)
    The Organic Chemistry Tutor11511
    $$\begin{array}{lll} f(3) = 4 & g(3) = -8 \\ f'(3) = -7 & g'(3) = 5 \\ [fg]'(3) = \mathop{?} \end{array}$$
    The Organic Chemistry Tutor11511