Standard Derivatives and the Basic Rules

Transcript

0:00Introduction

Hello everyone! Okay, a challenge first. Take x, times x minus one, times x minus two, and so on, all the way to x minus ten. Eleven brackets, multiplied together. Now find the derivative of this at x equals zero. Multiply out all eleven brackets? That could take an hour, right? By the end of this video, you will do it in ten seconds. Derivatives come up often in JEE, so today we build the toolkit: the standard derivatives and the basic rules. Let's begin.

0:35Standard derivatives: powers, exponentials, logs

First, the symbol on screen: d by d x. It means the derivative with respect to x, and it is not a division, okay? Now the power rule: d by d x of x to the power n equals n times x to the power n minus one. Bring the power down, then reduce the power by one. Clear? Quick check. Differentiate x cubed, plus five x squared, minus seven x, plus two. Go term by term. Three x squared, plus ten x, minus seven. Correct! Five x squared gives five times two x, that is ten x. The two is a constant, so its derivative is zero. Got it? Now, a to the power x: a fixed base, and x in the power. Can we use the power rule, and write x times a to the power x minus one? No, no, careful! The power rule needs a fixed power, and here the power is the variable. First, e to the power x: its derivative is e to the power x itself! And for a to the power x, we get a to the power x, times log a. The l n on screen is the natural log, log to the base e, and we read it as log. Okay? Then the logs. The derivative of log x is one over x. And for log of x to the base a, with the base written small below the log, we get one over, x times log a. Right? Last one here: the absolute value of x, written with two vertical bars. Its derivative is the absolute value of x, over x: plus one for positive x, minus one for negative x. At zero there is a sharp corner, so x is not equal to zero; that is the slashed equals sign. With me?

2:28The six trigonometric derivatives

Now trigonometry, six functions. The derivative of sine x is cos x, and the derivative of cos x is minus sine x. Watch that minus sign, okay? Next, tan x gives sec squared x; the small two after sec means sec x, the whole squared. And cot x gives minus cosec squared x. Right? Then sec x gives sec x tan x, and cosec x gives minus cosec x cot x. Got it? Now look at all six. Which three derivatives carry a minus sign? And what do their names share? Yes, exactly! Cos, cot and cosec, and all three names start with c o. So here is the topper's trick: every co function gets a minus sign. Clear?

3:18Product and quotient rules

Alright, now the rules. Take x squared, times sine x. The derivative of x squared is two x, and of sine x, cos x. So the derivative of the product is two x times cos x, right? No, no, wait! The derivative of a product is not the product of the derivatives. Here u and v are two functions of x, and the little dash, called prime, means the derivative. So, u v in brackets, prime, equals u prime v, plus u v prime. Differentiate one, keep the other, then swap, and add. Got it? So here, u is x squared and v is sine x. We get two x sine x, plus x squared cos x. Okay? And for a quotient, u over v: u prime v, minus u v prime, all over v squared. Careful with the order! Swap the two terms on top, and the answer flips its sign. So always start with u prime, the derivative of the top. With me?

4:23The chain rule: a function inside a function

Now, the chain rule. f of g of x is a function inside a function: g is inside, f is outside. Its derivative is f prime of g of x, times g prime of x. Differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside. Clear? Try sine of x squared. The outside is sine, which gives cos. So the answer is cos of x squared, right? Oh no, careful! We forgot the inside. The inside is x squared, and its derivative is two x. So we get cos of x squared, times two x; the dot means times. That is two x, times cos of x squared. This is the most common mistake here. Okay? One more: e to the power sine x. The outside, e to the power something, stays as it is; the inside, sine x, gives cos x. So, e to the power sine x, times cos x. Right?

5:28Example 1: quotient rule on (x squared + 1)/(x - 1)

Now, Example one. Differentiate x squared plus one, over x minus one. Pause the video, and try it first. Done? Okay, let's check. It is a quotient. The top, u, is x squared plus one, so u prime is two x. The bottom, v, is x minus one, so v prime is one. Clear? Now the formula: u prime v, minus u v prime, over v squared. So, two x times x minus one, minus x squared plus one, times one, all over x minus one, the whole squared. Keep the bracket around x squared plus one: the minus sign must reach both terms. Okay? Now simplify the top. What do you get? Two x squared minus two x, minus x squared, minus one: that is x squared minus two x minus one. So the derivative is x squared minus two x minus one, over x minus one, the whole squared. Got it?

6:31Example 2: f'(0) and f'(10) for x(x-1)(x-2)...(x-10)

Remember our challenge from the start? Let's finish it. f of x equals x, times x minus one, times x minus two, and so on up to x minus ten; the three dots mean and so on. Find f prime of zero, and f prime of ten. And no expanding! Pause the video, and try it. Done? Okay, let's check. Split f into two parts: x, times g of x. Here g of x is the product of the other ten brackets. Right? Now the product rule: f prime of x equals one times g of x, plus x times g prime of x. Put x equal to zero. The second term has the factor x, so it becomes zero, whatever g prime is! The arrow means so: f prime of zero is just g of zero. Beautiful! Now, g of zero is minus one, times minus two, and so on, up to minus ten. Ten negative numbers. So is the answer positive or negative? Positive, correct! Ten minus signs, an even number, so they cancel in pairs. What is left is one times two times three, up to ten: that is ten factorial, the exclamation mark on screen. Clear? And f prime of ten? Same idea: split off the bracket x minus ten. The other term keeps the factor x minus ten, which is zero at x equals ten. What is left is ten times nine, and so on down to one: ten factorial again. Ten seconds, not an hour. Challenge solved! With me?

8:14Quick revision: derivative formulas and the three rules

So, quick revision. One: learn the standard derivatives; the co functions get a minus sign. Two: the product rule and the quotient rule, with the quotient terms in that order. Three: the chain rule, the outside derivative times the inside derivative. Never forget the inside! Now go and try the practice problems on this page. All the best!