Use the alternate form of the derivative to the find the derivative of f(x)=x^4-3x^2+1 AT c=-2

Question

Use the alternate form of the derivative to the find the derivative of f(x)=x^4-3x^2+1    AT     c=-2

Answer 1

f'(c) = lim_x→c(f(x)-f(c))/(x-c)

 

So, f'(-2) = lim_x→-2[(x⁴-3x²+1-5)/(x+2)]

 

              = lim_x→-2[(x⁴-3x²-4)/(x+2)]

 

                     Divide x⁴-3x²-4 by x+2 to get x⁴-3x²-4 = (x+2)(x³-2x²+x-2)

 

So, f'(-2) = lim_x→-2[(x+2)(x³-2x²+x-2)/(x+2)]

 

              = lim_x→-2(x³-2x²+x-2) = -20

 

Answer 2

Hi John,

 

The alternative form of the derivative is the approximation of the derivative expressed as

 

lim   = f(x) - f(c)

x->c   -------------

            x - c

 

f(x) = x^4 - 3x^2 + 1 ,  c = -2

 

lim =  x^4 - 3x^2 + 1 - ((-2)^4 - 3*(-2)^2 + 1)
x->-2   -------------------------------------------------
                                 x - (-2)

 

lim    = x^4 - 3x^2 - 4

x->-2   --------------------------

                      x + 2

 

lim =    ( x^2 - 4) (x^2 + 1) 
x->-2  --------------------------
               x + 2

 

lim = ( x-2)(x+2) (x^2 + 1)
x->-2 --------------------------    x+2s cancel out
                x + 2

 

lim =  (x^2 + 1) (x - 2)
x->-2 

 

lim

x->-2 = -20