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Degree: Natural Sciences (Bachelors) - Durham University
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FIrslty to avoid confusion:
x = the variable x
X = multiplication.
When you differentiate an equation, the derivative (answer) is referred to as dy/dx.
For a polynomial as in this question you take each term individually and differentiate that.
Let's first do this for a generic term axb and we will then apply it to thisquestion:
The formula for the derivative of axb is baxb-1, which means you take the degree of x (whatever power it is being raised to, which in this case is b) and multiply it by the coefficient of x (which in this case in a) which provides the ba part of the solution. You then reduce the degree of x by one so it becomes xb-1 instead of just xb. Therefore when you put the two parts together you get the final derivative as baxb-1.
Now let's look at the first term of our equation : 5x3.
The degree here is 3 so the first thing you do is multiply 3 by the coefficient of x which is 5 giving you 3x5 which is 15.
Next, you reduce the degree of x by 1. It was x3 so now it is x3-1 which is x2.
Now you can put the two parts together to give the derivative of 5x3 being 15x2.
If you do the same thing for the next term, then you get (2x1/2)x2-1 which is x.
For the third time the derivative is quite nice, if you just have a number followed by x such as 3x as we have here, to get the derivative you just remove the x, meaning the derivative is 3.
For the final term of the equation we just have a number. Whenever you differentiate a number on its own, the derivative is 0.
So now the final thing left to do is put our answer together. You need to take each of our 4 derivatives and simply add them together.
This would give you: 15x2 + x + 3 + 0
Meaning the final answer dy/dx = 15x2 + x + 3.see more
The way to write a good hypothesis is to follow a 3 step proess.
1) Identify your variables and operationalise them.
2) Identify whether you are looking for a difference or a relationship.
3) Identify whether you are going to write a directional or non-directional hypothesis.
As long as your hypothesis includes these three things then it will be a strong statement.
Let's look at a specific example to see how we can do this:
The hypothesis we want to write is for a piece of research which is looking to see if the length of sleep impacts memory.
So let's go to step 1.
1) Our independent variable (which is the variable that we are able to change and manipulate) in this case is the length of sleep, and the dependent variable (which we cannot control but is what we measure) for this piece of research is memory. But now we need to operationalise them. Operationalising variables means explain how we measure the variable. So for example we could operationalise length of sleep to be 'people who slept more than 6 hours in comparison to people who slept less than 6 hours.' You often find that there are many ways to operationalise the dependent variable as something like memory can be measured in many ways. One way which you could operationalise the variable would be 'number of words correctly recalled from a list.'
So now we have both our operationalised variables, we can move on to step two.
2) We need to decide if we are looking for a difference or a relationship. A difference would be if we are directly comparing two things, whereas a relationship would be showing how one thing impacts another. If you are testing for a difference then your hypothesis will sound something like 'group A is more/less/different to group B' whereas if you are testing for a relationship you will say 'A increases/decreases/changes as B increases.' For this piece of research we are comparing people with more than 6 hours of sleep with those who had less than 6 hours of sleep so we are looking for a difference. This means our hypothesis will sound like 'people who sleep more than 6 hours will .... more/less/differently to people who slept less than 6 hours.'
Now we can move onto the final step of writing the hypothesis.
3) A hypothesis can be written as either directional (when you predict what the results will show, and so say 'A will be more than B or A will be less than B') or it can be non-directional (which is when you know that there will be a difference but do not know which one will be more or less so write 'A will be significantly different to B'). You can pick which type of hypothesis you want to write (unless the exam question specifies!) but for this example let's write a directional hypothesis. If we predict that more sleep will improve your memory we would write people who sleep more will have better memories than people who sleep less.
But now let's put everything together and write our final excellent hypothesis.
'People who sleep for more than 6 hours will recall more words correctly from a list than those who slept for less than 6 hours.see more