Differentiation

• Edexcel- C1 Differentiation
• AQA- C1 Differentiation
• OCR- C1 Differentiation

One of the new topics, we learn in A-Level Maths is Differentiation. It is one of the branches of Calculus, which is a major field in Mathematics, and almost is a useful application in loads of other fields.. Engineering, Economics, Physics and Chemistry. It is concerned with how one thing changes, as a result of another quantity changes. E.g. How displacement changes, as time changes (dd/ dt) would be the velocity.... We shall look for now at how y changes with respect to x (we call this dy/dx) = d (delta which means change)

We will look at curves, where the gradient is changing at each point on the curve. So dy/dx on each point is different, and not constant. First, i'll introduce you to a tangent. A tangent, is a straight line which touches a point on the curve, it only touches that point though.

The green line is the tangent to the curve( in black), this tangent only touches the yellow point on the curve. Note the gradient at the yellow point, is different to the rest of the points on the curve, because the gradient changes as x changes.

The derivative of a curve is the same as the dy/dx of a function e.g.

The derivative of x^3 = 3x^2
dy/dx (x^3) = 3x^2
This is the first derivative, if differentiate again, we would get the second derivative, again.. the third.. and so on..

dy / dx means differentiating y with respect to x. (what is happening to y, as x changes)

How to Differentiate

To differentiate a function, you reduce the power by 1, and multiply by the new power :

Function          Derivative
axⁿ                 anx<sup>n-1

e.g. x^2 
dy/dx = 2x


x^3 
dy/dx = 3x^2


* If we had to differentiate anything to the first power e.g. x , 3x, 5x... it would be 1,3 and 5 respectively. Why ? 
Because it is to the power 1, reducing the power to 0.. anything to the power of 0 equals 1 .. so we just multiply the coefficient of x by 1.. which is the same as taking away the x.

*Differentiating a number .. gives 0. Think about it, if draw a graph of say y = 5, the gradient is 0.

Other Notation 

A function can be written as y =... or f(x) = ..., if we have a function defined as f(x) =...., then the derivative of that is f'(x)=...

f(x) = 5x ... f '(x) = 5 (this is called f prime)

*For the first derivative we use one dash.. second derivative two..etc 
We can only differentiate functions in the form of</sup> axⁿ    , so if it looks any different, we have to rearrange to get in that form, using rules of indices.