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Rules for D ifferentiation

Rules for D ifferentiation. By Kimberly Low. Power Rule. The derivative of the function f(x) when f(x)= xⁿ is equal to: f(x)= xⁿ f’(x)= n(xⁿ⁻¹). Power Rule cont. SIMPLE EQUATIONS f(x)= x⁵ f(x)=3x⁷ f’(x)=5x⁴ f’(x)=21x⁶.

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Rules for D ifferentiation

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  1. Rules for Differentiation By Kimberly Low

  2. Power Rule • The derivative of the function f(x) when f(x)=xⁿ is equal to: • f(x)=xⁿ f’(x)= n(xⁿ⁻¹)

  3. Power Rule cont. SIMPLE EQUATIONS f(x)= x⁵ f(x)=3x⁷ f’(x)=5x⁴ f’(x)=21x⁶

  4. Power Rule cont. • More Complex Equations • f(x)=4x⁷-7x⁵+2x-1 f(x)=x⁸-2x⁶ • f’(x)=28x⁶-35x⁴+2 f’(x)=8x⁷-12x⁵

  5. Product Rule • h(x)=f(x)g(x) • Mathematical version: h’(x)=f(x)g’(x)+g(x)f’(x) • Written out version: The derivative of a function h(x) is equal to the first function [f(x)] times the derivative of the second function [g’(x)] plus the second function[g(x)] times the derivative of the first function [f’(x)].

  6. Product Rule cont. • h(x)= (2x-5)(5x+3) • h’(x)= (2x-5)(5)+(5x+3)(2) Use the rule! • h’(x)=10x-25+10x+6 Multiply it all out! • h’(x)=20x-19 Combine like terms! • *Make a box to help you stay organized!*

  7. Quotient Rule • h(x)= • Mathematical version: h’(x)= • Written differently: h’(x)= • Written out version: The derivative of a function h(x) is equal to the bottom function [f(x)] times the derivative of the top function [f’(x)] minus the top function [f(x)] times the derivative of the bottom function [g(x)], over the bottom function [g(x)] squared.

  8. Quotient Rule cont. • h(x)= • h’(x)= Use the rule! • h’(x)= Multiply! • h’(x)= Simplify!

  9. Derivative of a Number • f(x)=c f’(x)=0 • The derivative of any number, c, is equal to zero. • Example: • f(x)=4 f’(x)=0

  10. Chain Rule • f(g(x)) • Mathematical version: f(g(x))= f’(g(x))*g’(x) • Written out version: The derivative of the function [h(x)] is equal to the derivative of the first function [f(x)] with the second function intact [g(x)], times the derivative of the second function [g(x)].

  11. Chain Rule cont. • h(x) = • h’(x)= 2 (cos x) * (-sin x) • h’(x)= -2sin x cos x • h(x)= (3x-4)⁵ • h’(x)= 5(3x-4)⁴ * 3 • h’(x)= 15 (3x-4)⁴

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