We are given:
f(x)=-((x+1)(x-2))^(1/2)
f'(x)=d/dx[f(x)]
color(white)(f'(x))=d/dx[-((x+1)(x-2))^(1/2)]
color(white)(f'(x))=-d/dx[((x+1)(x-2))^(1/2)]
color(white)(f'(x))=-((x+1)(x-2))^(1/2-1)*1/2*d/dx[(x+1)(x-2)]
color(white)(f'(x))=-((x+1)(x-2))^(1/2-1)* 1/2*((x+1)d/dx[x-2]+(x-2)d/dx[x+1])
color(white)(f'(x))=-((x+1)(x-2))^(-1/2)/2*((x+1)(1)+(x-2)(1))
color(white)(f'(x))=-((x+1)(x-2))^(-1/2)/2*(x+1+x-2)
color(white)(f'(x))=-((x+1)(x-2))^(-1/2)/2*(2x-1)
color(white)(f'(x))=-((2x-1)((x+1)(x-2))^(-1/2))/2
f'(6)=-((2(6)-1)((6+1)(6-2))^(-1/2))/2=-(11sqrt(7))/28
However, for the normal, "gradient"=-1/(f'(x))=-1/(-(11sqrt(7))/28)=28/(11sqrt(7))
The normal will be in the form of y=mx+c,
m=28/(11sqrt(7))
x=6
y=-((6+1)(6-2))^(1/2)=-sqrt(7*4)=-2sqrt(7)
-2sqrt(7)=6*28/(11sqrt(7))+c
c=-2sqrt(7)-6*28/(11sqrt(7))
color(white)(c)=-(46sqrt(7))/11
y=(28x)/(11sqrt(7))-(46sqrt(7))/11
y=(4sqrt(7)x)/11-(46sqrt(7))/11
y=(4sqrt(7)x-46sqrt(7))/11
11y=4sqrt(7)x-46sqrt(7)
4sqrt(7)x-11y-46sqrt(7)=0
