Hey
I'd really appreciate some help with the following differntiation problems:

Here are my workings for the first part:
b^2 - 4ac = 0 (from formula)
4b^2 - 4(3a)(c) = 0
4b^2 = 12ac
b^2 = 3ac

For the second part, finding the coordinates, im not sure what i should do, any advice?

(ii) Using logarithms, or otherwise, differentiate:

y=(e^x^2\sqrt{SinX})/(2x + 1)^3

Ok, so I've been trying to do this, but I'm not sure where the logarithms part comes into it, or where to start, maybe a quotient rule?
Any help is appreciated.

"Here are my workings for the first part:
b^2 - 4ac = 0 (from formula)"

My first reaction was the same as FZ+: "where did you get that formula"? Then I realized that you are using the same letters to mean different things.

Given a cubic ax^{3}+ bx^{2}+ cx+ d, a turning point can only occur where the derivative: 3ax^{2}+ 2bx+ c= 0

That can be solved using the quadratic formula and, in order that there be exactly 1 turning point, there must be exactly one solution so "b^{2}- 4ac= 0" where a,b,c are now the coefficients of ax^{2}+ bx+ c. Putting in the values 3a and 2b for "a","b" gives you the rest:

"4b^2 - 4(3a)(c) = 0
4b^2 = 12ac
b^2 = 3ac"

BAD IDEA! Never use the same letters to represent different things! Or at least tell us you are doing that!

As for finding the turning point: 3ax^{2}+ 2bx+ c= 0 and
b^{2}= 3ac. Okay, then, again, from the quadratic formula,
since the discriminant is 0 x= -2b/(6a). You can plug that into the original cubic to find y (exactly what FZ+ suggested)

Like FZ+, I have no idea what "y=(e^x^2\sqrt{SinX})/(2x + 1)^3" is, in part because of that "\". My guess, like his, is that you meant
y=(e^{x^(x^2/sqrt(sin(x)))})/(2x+1)^3.

log(y)= x^(x^(2/sqrt(sin(x)))))-3log(2x+1) which should be easier to differentiate.

Ok, thanks for the help with the first part. I should probably have explained that the formula was taken from the quadratic formula x = [-b +/- squrt(b^2-4ac)]/2a, using the b^2-4ac = 0 to show that it has one real root.
I think the reason the question was asked using a,b,c,d was to confuse people trying to do it.

As for the log question, i didnt make a great job of typing it out, so ill try again:
Y = [{e^(x^2)}.sqrt{SinX}]/[(2x + 1)^3

Y = e to the power of x squared by the square root of sinX, divided by (2x + 1) to be cubed.
Thanks again for the help.