The Derivative

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle first. The slope of a line needs two points: rise over run, right? But for the tangent to a curve at a point, we know only one point on it. One point gives a rise of zero over a run of zero. Zero over zero! So how can a tangent have a slope at all? The answer is the derivative. It comes up often in JEE, and the rest of calculus builds on it. So, let's begin.

0:32The derivative from first principles

So, take a second point, a small distance h away, at x equals a plus h. The slope through these two points is rise over run: f of a plus h, minus f of a, all over h. The top is the change in f; the bottom, h, is the change in x. Clear? Now, we want the two points to come together. So, can we simply put h equal to zero in this fraction? No, no! Zero over zero again. Instead, we let h tend to zero and take the limit. Lim means limit, the arrow means tends to, and the little dash after f is called prime. So, f prime of a equals the limit as h tends to zero, of f of a plus h, minus f of a, over h. The derivative from first principles. Got it? But careful, there is a condition. f prime of a exists only if this limit exists, and is finite. No limit, or an infinite one, means no derivative there, okay? And the meaning? As h tends to zero, the second point slides into the first, and our line turns into the tangent. So, f prime of a is the slope of the tangent to y equals f of x, at the point a comma f of a. Puzzle solved! With me?

1:51Left-hand and right-hand derivatives

Now, just like limits, derivatives have two sides. First, the right-hand derivative, R f prime of a, with R for right. The small plus sign on the zero means h tends to zero from the right, so the second point is to the right of a. Same fraction, one side only. Right? Next, the left-hand derivative, L f prime of a. Look: h still tends to zero from the right, so the second point, a minus h, is on the left. But why is the bottom minus h, and not h? Yes, exactly! The bottom is the change in x: from a to a minus h, that is minus h. Oh no, careful here! Write plus h there, and every left-hand slope comes out with the wrong sign. That is a very common slip. Clear? Now the main rule. The double-headed arrow means if and only if. f is differentiable at a if and only if L f prime of a equals R f prime of a, and both are finite. Same idea as limits: the two sides must agree. Got it?

2:57Differentiable means continuous

Alright, now a big question. Suppose f prime of a exists. Must f be continuous at a? Yes, absolutely! Differentiable at a implies continuous at a. This arrow points one way, and it means implies. Okay? Why? Look at these lines. Write f of a plus h, minus f of a, as h times our fraction; the dot means times. As h tends to zero, the fraction goes to f prime of a, a finite number. So the product tends to zero times f prime of a, which is zero. So f of a plus h tends to f of a: continuity! Right? Now turn it around. Is every continuous function differentiable?

3:47Continuous does not mean differentiable

No, no! The converse fails. Look at the two graphs on screen. Left graph: y equals the absolute value of x, with vertical bars. No break, so it is continuous. But at the origin, a sharp corner: slope minus one on the left, slope one on the right. So L f prime is not equal to R f prime; the slashed equals sign means not equal. Not differentiable there. Clear? And the right graph: y equals the cube root of x; the small three on the root sign means cube root. Again, no break. But at the origin the curve stands straight up: the tangent is vertical, and the slope is infinite. Not finite, so not differentiable. With me? One last point. If f is discontinuous at a, it is never differentiable there. So, the topper's habit: check continuity first; if it fails, stop. But if it passes, the trap is to stop there. You must still compare L f prime and R f prime. Okay?

4:46Example 1: derivative of root x from first principles

Now, Example one. Find the derivative of root x, for x greater than zero, from first principles. Root x means the square root of x, the tick-shaped sign on screen. Pause the video, and try it first. Done? Okay, let's check. Same definition, at a general point x. The fraction is root of x plus h, minus root x, all over h. Put h equal to zero right away: what do we get? Zero over zero, correct! So we cannot substitute yet. The trick for square roots: multiply the top and the bottom by the same two roots with a plus in between, root of x plus h, plus root x. That is multiplying by one, so nothing changes. With me? Now the top is a difference of squares: x plus h, minus x. That is just h! Cancel this h with the h at the bottom, and we are left with one over, root of x plus h, plus root x. Right? No more zero over zero, so let h tend to zero. The bottom becomes root x plus root x, that is two root x. So, the derivative of root x is one over two root x, for x greater than zero. Got it?

6:07Example 2: is x sin(1/x) differentiable at x = 0?

Now a harder one, a type that comes up often. f of x equals x times sine of one over x, when x is not zero, and f of zero equals zero. Discuss its differentiability at zero. Pause the video, and try it. Done? Okay, let's check. Our habit: continuity first. The absolute value of sine is never more than one, so the absolute value of f of x is at most the absolute value of x. The sign on screen means less than or equal to. Clear? As x tends to zero, the absolute value of x tends to zero, so f of x is squeezed to zero. Limit zero, value zero: continuous at zero, okay? So, it is continuous. Then it must be differentiable too, right? No, no, wait! Remember the trap: check the derivative itself. Take the fraction at zero: f of zero plus h, minus f of zero, over h. f of h is h times sine of one over h, and f of zero is zero. Cancel h, and we get just sine of one over h. With me? Now think. As h tends to zero, one over h becomes huge. Does sine of one over h settle down to one value? No! It keeps swinging between minus one and one, faster and faster, and never settles. So this limit does not exist, and there is no derivative at zero. Right? So, the final answer: continuous, but not differentiable at zero. Continuity alone never proves differentiability. Got it?

7:54Quick revision: first principles, LHD = RHD, continuity

So, quick revision. One: f prime of a is the limit of this fraction as h tends to zero, when the limit exists and is finite. Two: differentiable if and only if the left-hand and right-hand derivatives are equal, and both finite. Three: differentiable implies continuous, but not the reverse: think of corners and vertical tangents. Now go and try the practice problems on this page. All the best!