18573 questions

### What is the chain rule and how does it work?

The chain rule is used when you're trying to differentiate a complicated function, and it allows you to split up the function into easier ones. The trick is spotting where to split the function, and what part you should substitute.  Say, for example, you're trying to find the derivative of:   y=(1+x)^2  . If you can replace a suitable function of x with a function of u, you would be left with y=f(u) and u=g(x), y and u being functions of u and x respectively. Now, if you differentiate y with respect to u, and u with repsect to x, you're left with the equation:  dy/dx = dy/du * du/dx  . On the RHS, you're left with dy/dx, as the du's cancel out, which is what you were after at the beginning. All that you would have to do is replace any remaining u with your function of x to find your desired derivative.  It's better understood through an example. Let's have a look at the example above, y=(1+x)^2. Take u=1+x, and youre left with y=u^2. Now, differentiate both fuctions, y=f(u) and u=g(x), and you're left with:  dy/du=2u  and  du/dx=1. Now, from the formula above, we know that dy/dx=dy/du * du/dx. Therefore:  dy/dx = 2u * 1 = 2u. Now all that's left to do is replace u with the function of x you originally took out. In this case, u=1+x, so:  dy/dx = 2(1+x) .    For a slightly harder example, try: y=1/(x^2+4x)^1/2 . Rearrange this to have y in terms of powers, similar to the first example: y=(x^2+4x)^-1/2 .  Now, take an appropriate function of x to equal u. In this example, it would be a good choice to take u=x^2+4x , leaving y=u^-1/2. Differentiating both of these functions gives:  dy/du=-1/2(u)^-3/2  and  du/dx=2x+4. We know that dy/dx=dy/du * du/dx, therefore:  dy/dx = -1/2(u)^-3/2 * (2x+4) . Replacing u with the function of x you took out gives:   dy/dx = -1/2(x^2+4x)^-3/2 * (2x+4)  .  Rearranging and simplifying this expression gives:  dy/dx = -(x+2) / (x^2+4x)^3/2  .
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Answered by Ethan, Maths tutor with MyTutor

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### What are the basic components of the immune system

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Answered by Daniel, who tutored Human Biology with MyTutor

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### How to transform graphs of functions?

So you've encountered something like: g(x) = | 4 f( |x + 1| - 3 ) + 5 | What a mess! And you are only given the graph of f(x). How do we go about sketching g(x)? As in any problem in mathematics, let's break it all down to a list of simple steps. We start with a graph of f(x). 1. Let's create a function f1(x) = f(x+1). You might already know that adding a number to the argument of a function is equivalent to 'sliding' its graph by that number to the left (it's quite simple, after all, you make an x=5 give f(5+1)=f(6) - which is to the right of f(5) we call that translation by a vector) 2. f1(x) is not quite like g(x) yet. Let's make f2(x) = f1( |x| ) = f( |x+1| ) . How does that affect the graph? As we know |x| = x, for positive x's and |x| = -x, for negative x's. So to sketch f2(x), we take the part of the graph of f1(x) which is to the right of the y-axis (where x is positive), and on the left half, we need to draw a mirror image of the right half. We're one step closer to g(x). 3. Next we need to include the number -3 in the argument. Let f3(x) = f2(x-3). Which means another shift along the x-axis. This time we move the graph of f2(x) to theright, by 'amount' 3.  4. Now we are 'outside' f3(x) - we are not going to play with the argument anymore. Which means that now we will change the graph in the y-axis only. The remaining steps include multiplying the function by 4. This means that everywhere on the graph, the function is 4 times farther away from the x-axis (from 0). So sketch f4(x) = 4*f3(x), 4 times taller and steeper. 5. Then there is the 5 left to add. If we add a number to the function, it means that it goes up in the y direction by 5, everywhere. So let's shift the graph upwards. f5(x) = f4(x) + 5 6. We are nearly done now. The last thing to do is the absolute value. Once again, for all positive values, it stays as is, and for all negative values it gets a minus sign. Thus everywhere the function f5(x) has values smaller than 0, we need to 'flip' it upside down (symmetry with respect to the x-axis). f6(x) = |f5(x)| g(x) = f6(x) And that is it. You now have an accurate depiction of g(x). It wasn't that complicated after all. You just needed to make incremental changes and step by step move 'from the inside out'. - See more at: https://www.mytutorweb.co.uk/tutors/secure/ta-yourexplanations.html#sthash.iwmqcICD.dpuf
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### What is division of labour within a cell?

'Division of labour' is a term that describes the specialised functions of cell organelles which come together to ensure the cell is capable of surviving as well as performing it's role in the body. For example, beta cells in the pancreas are responsible for releasing insulin into the bloodstream. In order for the beta cells to release insulin, there must be an effective 'division of labour' in the cell. Firstly, the gene fo insulin undergoes 'transcription' in the Nucleus and the subsequent mRNA molecule leaves the Nucleus through nuclear pores. Ribosomes attached to the Rough ER then synthesise the insulin from the mRNA template (a process known as translation). The insulin proteins are then transported in vesicles to the golgi appartus, where they may be modified slightly with the addition of a carbohydrate, or simply packaged into vesicles once again. The insulin is then navigated in the vesicles to the plasma membrane of the cell, where it is secreted via exocytosis. This example of division of labour shows how the Nucleus, ribosomes, Rough ER, vesicles and Golgi apparatus work in tandem to perform an essential metabolic function of the pancreas, however all organelles are involved in the division of labour within a cell as each organelle performs some kid of vital function needed for the survival of the cell as a whole.
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Answered by Ben, Biology tutor with MyTutor

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### How can I prepare to write an essay?

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Answered by Chloe, who has applied to tutor English with MyTutor

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### What qualifications and experience do you have at this level?

I have A* in Maths and Further Maths at A-level, and have tutored students in A-level maths for a year.
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Answered by Thomas, Maths tutor with MyTutor

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### Can you derive the Quadratic Formula?

For a general quadratic equation of the form ax2 + bx + c = 0 (1) We will solve for x via the ‘Completing the Square‘ method:   First we divide equation (1) through by a on both sides, yielding: x2 +(b/a)x + c/a = 0   Slightly rearranging, we then write (for tidiness): x2 + (b/a)x = -c/a (2)   Now the key behind completing the square is to try and write the x2 and x terms as the square of some quantity, which is close to:  (x + (b/2a))2 (3)   What we have done is half the coefﬁcient of the x term, as when the brackets are expanded out, you have a certain sum repeated twice, I will demonstrate in fuller detail now what I mean.   Expanding out (3), we have (x +(b/2a))(x+(b/2a)) = x2+ (b/2a)x + (b/2a)x + (b2/4a2) = x2 + (b/a)x + (b2/4a2) , which is almost identical to the left hand side of equation (2), except now we have the extra third term which is (b2/4a2)   We can rearrange this easily to show that x2+ (b/a)x =(x +(b/2a))2-(b2/4a2) Substituting this identity into equation (2), we now have (x +(b/2a))2 -(b2/4a2) = -c/a (4)   We now rearrange as follows: (x+(b/2a))2 = (b2/4a2) - (c/a) = (b2-4ac)/(4a2) , where in the third equality I have simply summed over a common denominator by multiplying the second term by 4a over the numerator and denominator.   Finally we take the square root of both sides, and rearrange to arrive at the quadratic formula we were looking for, as follows: x = (-b ± sqrt(b2-4ac))/2a   Note that we have two solutions to the quadratic equation, resulting from the existence of both positive and negative square roots, hence the ± sign. Also ‘sqrt' denotes taking the square root.   Corollary   Note that the solution of x depends on the quantity sqrt(b2-4ac), which is known as the Discriminant. This is important as it results with 3 distinct cases for the solution which are as follows:   1) b2 - 4ac > 0 , and so it gives one positive and one negative square root, leading to 2 unique solutions. 2) b2 -4ac = 0, which then yields exactly one solution, known as a repeated root, -b/2a . 3) b2 -4ac < 0, which cannot be square rooted within our common number system which we call the Real Numbers. If we extend our number system, to allow for square roots of negative numbers, we call these the Complex Numbers, which go far beyond the scope of the course.   If you want to ﬁnd out more, be sure to stay on till A Level and take Further Maths!
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Answered by Kamran, who tutored Maths with MyTutor

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### What is a stationary point and how do I find where they occur and distinguish between them?

A stationary point is simply a point on a graph where the derivative=0. Ie, the rate of change of the curve at this point is 0 and therefore it is neither increasing or decreasing at this point.  There are three types you need to know about: 1) A maximum: Here the derivative =0 and the second derivative <0. 2) A minimum: Here the derivative =0 and the second derivative >0 3) A point of inflection: Here the derivative and the second derivative =0 Note, the second derivative means the derivative of the first derivative! General solution: Suppose y=f(x) and dy/dx=f'(x) If at a point, say c, f'(c)=0 then there is a stationary point at this value of x. Differentiate f'(x) to get the second derivative. Plug in the value of c again and if the solution is.. 0 - Point of inflection Positive - Minimum turning point Negative - Maximum turning point Example y = x3 - 6x2 + 9x - 4 Find any stationary points and determine their nature. Solution  dy/dx = 3x2- 12x + 9 At a stationary point, dy/dx=0 So 3x2- 12x + 9 = 0 3(x2- 4x + 3) = 0   (x - 3)(x - 1) = 0 So stationary point at x = 3 and x = 1. Now, to determine the nature of these.. f''(x) = 6x - 12 f''(3) = 18 - 12 = 6 therefore minimum turning point at x = 3 f''(1) = 6 - 12 = -6 therefore maximum turning point at x = 1
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Answered by Anna, Maths tutor with MyTutor

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