Chain rule

Lessons

Chain Rule
if: $y = \;f\left( {\;\;\;\;\;\;\;} \right)$
then: $\frac{{dy}}{{{d}x}} = f'\left( {\;\;\;\;\;\;\;} \right)\cdot\frac{{d}}{{{d}x}}\left( {\;\;\;\;\;\;\;\;} \right)$

Differential Rules

• Introduction
  Introduction to Chain Rule
  • “bracket technique” explained!
  • exercise: $\frac{d}{dx}x^{10}$ VS. $\frac{d}{dx}(x^5+4x^3-6x+8)^{10}$

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• 1.
  Differentiate: Polynomial Functions
  $\frac{d}{dx} (2x-1)^3$

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• 2.
  Differentiate: Rational Functions
  a)

  $\frac{d}{dx} \frac{1}{(4x^3+7)^{10}}$
  
  
  b)

  $\frac{d}{dx}- \frac{5}{\sin ^2x}$
  
  

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• 3.
  Differentiate: Radical Functions
  a)

  $\frac{d}{dx} \sqrt{x^3+4x^2-9}$
  
  
  b)

  $\frac{d}{dx} {^3}\sqrt{(x^2+5)^7}$
  
  
  c)

  $\frac{d}{dx} \frac{1}{{^3}\sqrt{6x^4-x}}$
  
  
  d)

  $\frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}}$
  
  
  e)

  $\frac{d}{dx} {^3}\sqrt{\ln x}$
  
  

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• 4.
  Differentiate: Trigonometric Functions
  a)

  Differentiate: $y= \sin ^4x$

  VS.

  $y=\sin (x^4)$
  
  
  b)

  $\frac{d}{dx} \tan (\cos e^{5x^2})$
  
  
  c)

  $\frac{d}{d \theta} \sin (\cos (\tan \theta))$
  
  

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• 5.
  Differentiate: Exponential Functions
  a)

  $\frac{d}{dx} e^{\tan x}$
  
  
  b)

  $\frac{d}{dx} e^{\csc 5x^2}$
  
  
  c)

  $\frac{d}{dx} 2^{\sin x}$
  
  
  d)

  $\frac{d}{dx} 5^{2^{{x}^3}}$
  
  

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• 6.
  Differentiate: Logarithmic Functions
  a)

  $\frac{d}{dx} \ln x^{100}$

  VS.

  $\frac{d}{dx} (\ln x)^{100}$
  
  
  b)

  $\frac{d}{dx} \log_{2}{x^3}$
  
  

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