What does differentiation actually mean?

When you differentiate an equation, you're finding the gradient of its graph.<o:p></o:p>

For example, if you differentiate the equation y = x² you get a solution dy/dx = 2x<o:p></o:p>

This means that if you drew a line at a tangent to the curve of x² at any point, and found the equation of that line in the form y = mx + c (where m is the gradient of the line, and c is the intercept) then the m value of that line would be 2x (with the x value at that point).<o:p></o:p>

This makes sense; when y = 0 , the gradient of the curve is 0, and as x increases, y increases by 2x for every 1 that x increases by. Looking at the graph of x², we can see that y does get bigger and increases more rapidly as x gets bigger; the slope or "differential" of the curve gets steeper.<o:p></o:p>

But why is this useful?<o:p></o:p>

Because the differential tells us the rate of change of x with y. It tells us how much y is changing as x changes, so it helps us to understand the relationship between x and y.<o:p></o:p>

For example, imagine you're running a chemical reaction, with product "B". You want to make as much "B" as possible from your input "A". You know the relationship between A and B is given by B = 3A² - 12A . <o:p></o:p>

Then you can find the differential dB/dA = 6A - 12 , which is positive so long as A is bigger than 2. As the differential is positive, then we know B is increasing, and we can see its increasing faster than A, as for every unit A increases, B increases by 6A-12. <o:p></o:p>

Therefore, you know that you want to make B in big batches, as you get more B out for every unit of A you put in. You also know that you definitely don’t want to make B with less than 2 units of A.<o:p></o:p>

While this is a simple example, differentiation can be used on more complex equations in maths, physics, biology and chemistry to solve all kinds of problems.<o:p></o:p>