Calculus 2

General Polynomial Function

Consider (an - bn)

divided by (a - b)

The following shows this.

Now consider the general polynomial

y = anxn + an-1xn-1 + an-2xn-2 + .... an-1xn-1 + a2x2 + + a0x0

Now if point A is (x,y) and point B is (x+∂x,y+∂y)

Then

∂y = an((x+∂x)n-xn) + an-1((x+∂x)n-1-xn-1) + ..... + a2((x+∂x)2-x2) + a1((x+∂x)1-x1) + a0((x+∂x)0-x0)

∂x = x + ∂x - x = ∂x

Consider the general rth term in the above polynomial.

    ar((x+∂x)r-xr)

which is to be divided by ∂x = (x+∂x) - x

this is the same form as shown in figure (an-bn) / (a-b) above with a=(x+∂x) and b=x and n=r. So for the rth term

∂y/∂x = (x+∂x)r-1 + (x+∂x)r-2x + (x+∂x)r-3x2 + (x+∂x)r-4x3 + ....

   ..... + (x+∂x)2xr-3+ (x+∂x)xr-2 + xr-1

As ∂x tends to 0 all terms that are multiplied by ∂x also tend to 0 and so can be ignored.

Lim ∂y/∂x  = dy/dx

 ∂x → 0

and so for the rth term

dy/dx = xr-1 + xr-2x + xr-3x2 + .... + x2xr-3 + xxr-2 + xr-1

      = rxr-1

Note: in the above xr-1 appears twice, making the sum of terms r and not r-1.

Real Example y=x5