Chain rule

  1. Introduction to Chain Rule
    • "bracket technique" explained!
    exercise: ddxx10\frac{d}{dx}x^{10} VS. ddx(x5+4x36x+8)10\frac{d}{dx}(x^5+4x^3-6x+8)^{10}
  1. Differentiate: Polynomial Functions
    ddx(2x1)3 \frac{d}{dx} (2x-1)^3
    1. Differentiate: Rational Functions
      1. ddx1(4x3+7)10 \frac{d}{dx} \frac{1}{(4x^3+7)^{10}}
      2. ddx5sin2x \frac{d}{dx}- \frac{5}{\sin ^2x}
    2. Differentiate: Radical Functions
      1. ddxx3+4x29\frac{d}{dx} \sqrt{x^3+4x^2-9}
      2. ddx3(x2+5)7 \frac{d}{dx} {^3}\sqrt{(x^2+5)^7}
      3. ddx136x4x\frac{d}{dx} \frac{1}{{^3}\sqrt{6x^4-x}}
      4. ddxx+x+x \frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}}
      5. ddx3lnx \frac{d}{dx} {^3}\sqrt{\ln x}
    3. Differentiate: Trigonometric Functions
      1. Differentiate: y=sin4xy= \sin ^4x
        y=sin(x4)y=\sin (x^4)
      2. ddxtan(cose5x2) \frac{d}{dx} \tan (\cos e^{5x^2})
      3. ddθsin(cos(tanθ))\frac{d}{d \theta} \sin (\cos (\tan \theta))
    4. Differentiate: Exponential Functions
      1. ddxetanx\frac{d}{dx} e^{\tan x}
      2. ddxecsc5x2\frac{d}{dx} e^{\csc 5x^2}
      3. ddx2sinx\frac{d}{dx} 2^{\sin x}
      4. ddx52x3\frac{d}{dx} 5^{2^{{x}^3}}
    5. Differentiate: Logarithmic Functions
      1. ddxlnx100 \frac{d}{dx} \ln x^{100}
        ddx(lnx)100\frac{d}{dx} (\ln x)^{100}
      2. ddxlog2x3 \frac{d}{dx} \log_{2}{x^3}
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    Topic Notes
    Chain Rule appears everywhere in the world of differential calculus. Whenever we are finding the derivative of a function, be it a composite function or not, we are in fact using the Chain Rule. In this section, we will learn about the concept, the definition and the application of the Chain Rule, as well as a secret trick – "The Bracket Technique".
    Chain Rule
    if: y=  f(              )y = \;f\left( {\;\;\;\;\;\;\;} \right)
    then: dydx=f(              )ddx(                )\frac{{dy}}{{{d}x}} = f'\left( {\;\;\;\;\;\;\;} \right)\cdot\frac{{d}}{{{d}x}}\left( {\;\;\;\;\;\;\;\;} \right)

    Differential Rules
    table of chain rule applications on various functions 1
    table of chain rule applications on various functions 2