# Differentiation Basics

by Batool Akmal

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00:02 Welcome to calculus methods.

00:04 Now previously we've looked extensively in differentiation from first principles.

00:09 And we've also now observed a much faster way of differentiating.

00:14 In this lecture, we are now going to take faster method and we are actually going to start looking at some application of differentiation.

00:21 So, we're going to go over the differentiation basics today.

00:28 We'll look at the standard method of differentiation which is the faster method that we've just spoken about.

00:34 And we'll practice it with few more questions.

00:37 And then we're going to move on to more exciting bits which are the applications of it.

00:41 So we're going to look at gradients of a particular point on a curve.

00:45 How to find equations of tangents.

00:47 How to find equations of normals.

00:49 And again this is really useful for courses like medicine where you are actually looking at gradients of curves.

00:56 So that you can work out and the steepness or the speed, that things are growing or decaying with.

01:03 So previously we have found the gradient of a function like this 3x(squared) + 5x-1 using the definition of first principles.

01:12 We applied the really long formula and came to the final answer.

01:16 Where we made the delta X or the limit of delta X go to 0.

01:21 But thankfully to modern calculus we now have much faster ways of doing this.

01:25 And we discuss that faster way previously where you bring the power down and multiply it, and you decrease the power by 1.

01:32 Remember what we said that any constants at the end go to 0 or they just disappear.

01:40 As a little recap we will have a look at the general rule once again.

01:44 So if you have a function y = x to the power of n.

01:49 You bring the power down as the multiple of X to the power of n as we do here.

01:55 And then you decrease the power by 1.

01:58 So the answer will then be nx to the n - 1.

02:02 Remember we've discussed this before just by observation, looking at differentiation from first principles.

02:08 Also we've done some questions on this.

02:11 And we've also derived it and proved to ourselves that this is true using differentiation for first principles.

02:18 And as mathematicians, we like proofs because that shows us that what we're doing is the truth.

The lecture Differentiation Basics by Batool Akmal is from the course Calculus Methods: Differentiation.

### Included Quiz Questions

1. A point
2. An interval
3. Both in an interval and point
1. 0
2. 1
3. -1
4. x
5. 1/x

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