#color(blue)("Finding the gradient at any point")#
#color(brown)("Using shortcut method - very fast")#
Set #" "y=x^3#
#d/(dx) (x^3) -> nx^(n-1) = 3x^2 #
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Using first principles")#
Increment #x# by the minute amount of #delta x#
As #x# has changed then #y# will also change.
Let the minute change in #y# be #delta y#
Thus #y=x^3" "-> "y + delta y=(x + delta x)^3#
#=>y + delta y = (x + delta x)(x^2+2xdelta x +(delta x )^2)#
'..................................................................................
Consider:#" " x(x^2+2xdelta x +(delta x )^2)#
#x^3+2x^2(deltax)+x(deltax)^2#
'...................................................................................
Consider:#" "+deltax(x^2+2xdeltax+(deltax)^2)#
#x^2(delyax)+2x(deltax)^2 + (deltax)^3#
'.........................................................................
Putting it all together
#=>y + delta y= x^3+2x^2(deltax)+x(deltax)^2+x^2(deltax)+2x(deltax)^2 + (deltax)^3#
#=>y+deltay= x^3+3x^2(deltax)+3x(deltax)^2+(deltax)^3#
Subtract the original equation of #y=x^3#
#deltay=3x^2(deltax)+3x(deltax)^2+(deltax)^3#
Divide throughout be #delta x#
#(deltay)/(deltax )= 3x^2(deltax)/(deltax)+3x(deltax)^2/(deltax)+(deltax)^3/(deltax)#
#(deltay)/(deltax)=(3x^2xx1)+(3x xx deltax) +(deltax)^2#
#lim_(deltax->0) (deltay)/(deltax)= (dy)/(dx)=3x^2+(3xx0)+(0^2)#
#=>(dy)/(dx)=3x^2" "->"Tangent gradient"#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
At #x=2" "(dy)/(dx)-> m =3(2)^2 = 12#
So #color(brown)(y=mx+c)" "color(blue)(->" "y=12x+c)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
From the original equation of #y=x^3# at #x=2" ; "y=(2)^3=8#
So the tangent passes through the point #(x,y)->(2,8)#
Thus #y=12x+c->8=12(2)+c#
#=> c=-16#
#" "underline(bar(color(red)(|=> y=12x-16|)))#
