Differentiation basics: What is it?

Note: sorry for the lump of text, the format of the site will not allow paragraphs unfortunately... when I put // it means new section or new paragraph, sorry again for the inconvenience caused.//

Differentiation, what is it?:
Differentials are used to find the rate of change of one variable relative to another. We write a differential as dy/dx when y is a function in terms of x or when y is the independent variable and x is the dependent variable. // Graphical applications: If given the equation of a line in the form y=f(x) we can use differentiation to find the rate of change of the line at any given x co-ordinate, this is also known as the gradient of the line. // Generic Rules:If we say the y is a general function of x, y=ax^(b), (where a is a simple constant, or number in front of the function and ^b represents the numerical power applied to x, like squared or cubed if x is 2 or 3) then we can express the differential of the equation using this simple form which will work for all real values: dy/dx= abx^(b-1) where a,b and x are all multiplied together but we remove the multiplication signs for ease of viewing. The process put into words is that we bring the original power down from the x to the front, which means we are multiplying through by it, and then we are reducing the power by a value of 1 to create the new power for the differential, the constant a is unaffected by this process so remains at the front, if a times b can be simplified ( when they are both number or both in terms of the same unknowns) then you simply write out the differential again with the simplified product of a and b, called a coefficient. // Examples: If y=6x^4 what is dy/dx? we can see that a=6 and b=4 from our generic rule so, dy/dx=6(times)4(times)x^(4-1) therfore: dy/dx=24x^3 // Another example is if y=x^(-2) so a=1 and b=-2, just because b is negative doesn't change our rule, it just means that you have to be aware its negative and be a bit more cautious. dy/dx=1(times)-2(times)x^(-2-1) therefore: dy/dx=-2x^(-3)// Note: if the power is negative then putting it in brackets can help make sure you see it is negative, it reduces chances of mistakes. Ultimately its up to you if you want to put brackets around it, but I would suggest it just to avoid any silly mistakes :) //Hopefully that helps you understand the basics of differentiation, its purpose and also its origin. If you have any questions or problems at all please don't hesitate to ask me, additionally if there is a particular question you need help with or want a worked solution for just send it to me and I will get that done for you :)

That's the bare basics of differentiation, in the next few lectures we will go through for advanced forms of differentiation rules such as the product rule, the chain rule and looking at differentiation of trigonometric functions!