Differentiation

What is differentiation?

Say you have any polynomial y=f(x), and you go to a random point and draw a tangent at that point. You can then figure out the gradient of the tangent to find the gradient of the polynomial at that point. You would think that we'd have to do this procedure whenever we want to know the gradient at a certain point of y=f(x), but we can do better. We can do differentiaion, and I'll show you how to do it shortly. The notation of a differentiated funcion is either f'(x) or ^dy⁄_dx. Differentiation is the rate of change of a polynomial, as in how much does it increase/decrease.

How to Differentiate

If we take the general polynomial, y=axⁿ, then to differentiate we need to multiply the coefficent 'a' by the power 'n', and then decrease the power by 1, so we get anx^n-1. An example of this that I use is f(x)=x², as f'(x)=2x. I like this as it's simple. More complex polynomials can also be differentiated, for example y=4x³-8x²+2x-12, as all you have to do is differentiate them seperately to get ^dy⁄_dx=12x²-16x+2. Note that the +12 dissapears as its power is 0, and anything times 0 is 0.

Applications of differentiation

There are many uses for differentiation. I'll go over some of the ones you need to know

Finding the gradient

Finding the gradient at any point on a polynomial is easy; all you need to do is differentiate that funcion and then substitute in the x-value. For example, the gradient on the curve x³ when x is 5 is basically asking what is 3x² when x is 5, the answer is 75.

Finding the equations of tangents and normals at a point

Following on from the previous point, we use the same method to find the gradient at the point, and then we can use that value to find the gradient of a tangent or normal at that point. If its a tangent, then the gradient stays the same, but if it is a normal, then we take the negative reciprocal of that value. To get the equation of the line, put the x-coordinate into the original equation to get the y-coordinate (not the gradient), and then use the y-y₁=m(x-x₁) method to get the equation. See here if you cannot remember how to do it.

Finding maxima and minima

Maxima and minima are points on the polynomial where the gradient is 0 and they are locally either the lowest (minima) or the highest (maxima). To find them, firstly find f'(x) and then set f'(x) equal to 0 and then solve to find x. Then, to tell if they are either maxima or minima you can use either of these methods:

• Firstly, differentiate f'(x) again to get f''(x), or the second derivative. Then, put the x value in where you think there is a maxima or minima:
• If f''(x) is less than 0 then there is a local maximum at x
• If f''(x) is greater than 0 then there is a local minima at x
• If f''(x) equals 0 then you have to use the other method
• The second method is to go back to the original equation, and at where you think there is a maxima or minima, find the y values when x is slightly increased and slightly decreased. If the values around x are higher, then it is a local minima, and if they are lower, then it is a local maxima.

Increasing and Decreasing Functions

A function is said to be increasing if when the x values increase, so do the y values. Hence, we can say if the gradient is positive then the function is increasing. The opposite is also true for decreasing functions (gradient is negative)