Math Is Fun Forum

f(x) = x²/(1+x²)
f(x) = x²(1+x²)-¹
Now, use the uv method:- u'v + uv'
f'(x) = 2x(1+x²)-¹ + x²(-1)(1+x²)-²(2x)

This is because you didn't want to use u/v = (u'v - uv')/v²

In a parallelogram,
opposite sides are parallel, and equal; opposite angles are equal, and the diagonals (lines inside that intersect) bisect each other.
When you know the length of the diagonals, half of them would be the sides of a traingle they form with one of the sides of the parallelogram.

Use the theorems that (i) when two lines intersect each other, the vertically opposite angles are equal and (ii) sum of the total angles is equal to 360 degrees. This way, all the angles can be known.

Now, use the Cosine Theorem
a² = b² + c² - 2bcCosA
(where A, B, C are three angles of a traingle and a,b,c are the three sides opposite to angles A,B,C respectively)
for knowing the third side of the triangle, which forms a side of the parallelogram. Following this method, the adjacent side too can be found! Since opposite sides of a parallelogram are equal, we know all the four sides!

I know this reply is long, and may not be of much help; that's because some Mathematical problems are difficult to explain without a diagram!

Welcome to the forum, John!
Your posts and replies are interesting, keep posting

Thanks Rora, we miss you in this forum!

Mikau, Ahgua.....I am sorry, yesterday, when I was
trying to copy the diagram, I saw two same posts
(maybe, I thought I saw) of the diagram, to simplify,
I tried to delete one, and the entire post and got deleted.
I shall be much much more careful in the future,
and thanks for posting it before I requested you!

Nice question, brilliant reasoning.

Problem #k+4

A polyhedron has 12 faces(sides) and 18 edges. How many vertices does it have?

cosecA + cotA = 3
To find cosecA - cotA,
one should remember,
cosec²A - cot²A = 1
From this, we get
(cosecA+ cotA)(cosecA - cotA)=1
Given cosecA + cotA = 3,
3(cosecA-cotA)=1
cosecA-cotA=1/3

cosecA + cotA + cosecA-cotA = 3+1/3 = 4/3
2cosecA = 4/3
cosecA=2/3
sinA=3/2 funny

Problem #k+3

A cube of dimensions 10cm x 10cm x 10cm is made up of smaller cubes of dimensions 1cm x 1cm x 1cm. If the outermost layer of smaller cubes are taken off, how many smaller cubes would be left?

I expected this would happen.......
A, T, G, C are NOT
Adenine, Thymine, Guanine and Cytosine.

Problem #k+2

I have a certain number of eggs in my store.
If you divide the number of eggs by 2 there will be one egg left.
If you divide the number of eggs by 3 there will be two eggs left.
If you divide the number of eggs by 4 there will be three eggs left.
If you divide the number of eggs by 5 there will be four eggs left.
If you divide the number of eggs by 6 there will be five eggs left.
If you divide the number of eggs by 7 there will be six eggs left.
If you divide the number of eggs by 8 there will be seven eggs left.
If you divide the number of eggs by 9 there will be eight eggs left.
If you divide the number of eggs by 10 there will be nine eggs left.

If you divide the Number of eggs by 11 there will be NO EGGS left!
What is the minimum number of eggs I can have in my store?

Good wishes to all awaiting GCSE results