Let's have a look at our second question.
We are now looking at a function that is being,
has a number and has a function dividing it.
Let's just discuss the kind of things we can do here
so we have y equals to 1 over x squared, plus 2x, plus 1.
Now you are often able to bring the whole function up if you ever wanted to
and just use the product rule. However, in this case we'll just stick to the quotient rule
because it's probably the easiest way to do it.
You can say that the top function is u and the bottom function is v.
Remember we apply the quotient rule when we notice that two things are dividing.
Also recall the rule for differentiation so dy/dx, according to the quotient rule,
is vdu/dx minus udv/dx, over v squared, so we need all of these components to put them into this formula.
If we look at u now we have 1, so it's not actually a function of x
but we'll see what we can do with it. V is x squared, plus 2x, plus 1.
When you differentiate 1 or any number, it'll just go to zero.
Remember what we just said constants just disappear. They just go to zero.
So this might look a bit complicated but in fact, it makes the entire solution a lot easier.
Let's continue. We now differentiate v dv/dx or v dash, that gives you 2x plus 2
and obviously that one just disappears. We can now apply the quotient rule that says
vdudx minus udvdx divided by v squared so we have dy/dx.
When I multiply v with u that gives me zero multiplied by x squared plus 2x plus 1
minus and we now have 1 multiplied by 2x plus 2 all over v squared
so x squared, plus 2x, plus 1, all squared at the bottom.
So the good thing that happens here, anything multiplied with zero just becomes zero
so that term will just disappear leaving us with just minus 2x plus 2
because 1 doesn't bother us, over x squared, plus 2x, plus 1, all squared.
I wouldn't bother expanding the bottom denominator here
because it is factorize and it is easier so keep your lives simpler
rather than make the more complicated. So the derivative of this function y
equals to 1 over x squared, plus 2x, plus 1 is this function here
and again remember we can put values into it to find gradients at specific points.