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# Core 3 differentiation question

1. Show that there is just one point P, on the curve y = x / 1+x where the gradient is 7. find the coordinates of P.

I used the product and chain rule to get:
dy/dx = 3x(x-1)^2 +1(x-1)^3

Then I took (x-1)^2 out of both terms and got:
dy/dx = (x-1)^2(4x-1)

Then I put it equal to 7:
7 = (x-1)^2(4x-1)

But I'm having trouble solving. How can I solve this to find one value of x? Is this equation even correct?

2. (Original post by Bhansongal98)
Show that there is just one point P, on the curve y = x / 1+x where the gradient is 7. find the coordinates of P.

I used the product and chain rule to get:
dy/dx = 3x(x-1)^2 +1(x-1)^3

Then I took (x-1)^2 out of both terms and got:
dy/dx = (x-1)^2(4x-1)

Then I put it equal to 7:
7 = (x-1)^2(4x-1)

But I'm having trouble solving. How can I solve this to find one value of x? Is this equation even correct?

Your derivative doesn't seem to match the function you've given.

I suspect you meant

In which case your derivative is correct.

One can perhaps see by inspection that a certain value of x will give the desired result.
Spoiler:
Show

x=2
To show that this is the only value, expand the brackets to get a cubic. Use polynomial/long division to remove the linear factor (the root you've found), and you're left with a quadratic, which you need to show has no roots, and hence there is only one value of x satisfying that equation.

There may be an easier method I've totally missed.

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