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y=f(x)=4x3+3x-2

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2 Answers

 f( X ) = 4 X^3 + 3X -2
 
  
[ f ( X + h ) - f (X ) ] / ( X + h - X) =
h → 0
 
 
 { [ 4 ( X + h ) ^3 + 3 ( X +h) - 2 ] - (4 X^3 + 3X - 2) } / h =
h→0
    [4 X^3 + 12X^2 h + 12 h^2 X +3X + 3h + h^3 -2 ) - 4X^3 -3X +2 ] /h =
 
   ( 12 X^2 h + 12h^2X +h^2+ 3h ) / h =
h →0
       ( 12 X^2 +12hX + h +3) =
  h →0
          12X^2 + 3
 
“y = f(x) = 4x^3 + 3x - 2
Find the derivative dy/dx using f(x+Δx)-f(x)/(Δx).”

f(x+Δx) = 4(x+Δx)^3 + 3(x+Δx) - 2

Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1

(x+Δx)^3 = 1 x^3 + 3 x^2(Δx) + 3 x(Δx)^2 + 1 (Δx)^3

f(x+Δx) = 4(x^3 + 3 x^2(Δx) + 3 x(Δx)^2 + (Δx)^3) + 3x + 3Δx - 2

f(x+Δx) = 4x^3 + 12 x^2(Δx) + 12 x(Δx)^2 + 4(Δx)^3 + 3x + 3Δx - 2

-f(x) = -4x^3 - 3x + 2

f(x+Δx)–f(x) = 12x^2(Δx) + 12x(Δx)^2 + 4(Δx)^3 + 3Δx

(f(x+Δx)–f(x))/(Δx) = 12x^2 + 12x(Δx) + 4(Δx)^2 + 3

Δx → 0 ==> (f(x+Δx)–f(x))/(Δx) → 12x^2 + 3 = dy/dx

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