power rule:
#(deltay)/(deltax) x^n = nx^(n-1)#
when an expression #x^n# is differentiated, the result will have a degree #1# less than that of the original.
e.g. #(deltay)/(deltax) x^7 = 7x^6#
if this rule is used for each stage of the derivative, the #4#th derivative will have a degree #4# less than that of the original expression.
#(deltay)/(deltax) x^7 = 7x^6#
#(delta^2y)/(deltax^2) x^7 = 42x^5#
#(delta^3y)/(deltax^3) x^7 = 210x^4#
#(delta^4y)/(deltax^4) x^7 = 840x^3#
the fourth derivative of #x^7# has a degree #4# lower than #x^7# does.
#(delta^4y)/(deltax^4) x^7 = 840x^(7-4)#
if a polynomial is #7#th degree, the highest degree is #7#.
the highest degree of the fourth derivative would therefore be #7-4#, which is #3#.
