Implicit and Parametric Differentiation

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Take the circle x squared plus y squared equals twenty five. The point three, four lies on it, right? Nine plus sixteen is twenty five. Now, how would you find the slope of the tangent at that point? Usually we want y alone first. But here, y equals plus or minus the square root of twenty five minus x squared. Two pieces, and a square root. Messy, isn't it? Today we find slopes without ever making y alone. Two tools: implicit differentiation and parametric differentiation. They come up often in JEE, and they save a lot of time. So, let's begin.

0:44Implicit: y is still a function of x

So, here is the idea of implicit differentiation. We do not make y alone. We simply remember that y is still a function of x, and we differentiate both sides of the equation with respect to x. Okay? Now a quick question. This symbol, d by d x, means: differentiate with respect to x. So, what is d by d x of y squared? The small raised two means squared. Many students say two y, straight away. Is that right? No, no, wait! That is the trap. y depends on x, so we need the chain rule. First two y, then times the derivative of y itself, d y by d x. So d by d x of y squared equals two y times d y by d x. We also write d y by d x as y prime; the small dash is called prime. Got it? After that, it is just algebra. Collect the y prime terms on one side, and solve for y prime. Clear?

1:45Shortcut: dy/dx = -Fx/Fy for F(x, y) = 0

Now, a shortcut. Bring everything to one side, so the curve reads capital F of x, y equals zero. Then d y by d x equals minus F sub x, over F sub y. The small letters written below are subscripts. And F sub y must not be zero; the equals sign with a slash means not equal to. Okay? So what do these subscripts mean? F sub x: differentiate F with respect to x, holding y constant. And F sub y: differentiate with respect to y, holding x constant. Right? Let's try it on a curve from your solved examples, with everything on one side: F equals x cubed plus y cubed minus three a x y. The small raised three means cubed. Tell me, what are F sub x and F sub y? F sub x is three x squared minus three a y. Yes! The y is just a constant here. And F sub y is three y squared minus three a x. With me? So d y by d x is minus this, over this. Take the minus sign to the top, cancel the threes, and we get a y minus x squared, over y squared minus a x. One line! But never drop the minus sign in front. That is the usual slip, okay?

3:10Parametric: x and y both depend on t

Now the second tool. Sometimes a curve is given through a third letter, t, called the parameter. x is a function of t, and y is a function of t. Now think. How do we get d y by d x from these two? So x equals x of t, and y equals y of t. Differentiate each one with respect to t, then divide. d y by d x equals d y by d t, over d x by d t. It looks as if the d t cancels. Of course, d x by d t must not be zero. And careful: y on top, x at the bottom. Flipping this fraction is a very common slip. Clear? Let's try one from your solved examples. x equals a cos t, and y equals b sin t. What are d x by d t and d y by d t? d x by d t is minus a sin t, and d y by d t is b cos t. Correct! Watch the minus sign: the derivative of cos is minus sin. Okay? Now divide, y part on top: b cos t over minus a sin t. Cos over sin is cot, so this is minus b over a, times cot t. Done in two lines. Got it?

4:27Topper's trick: an expression that never ends

Now a trick toppers use. Look at this: y equals the square root of x, plus the square root of x, plus the square root of x, and so on. The three dots mean it goes on forever. How can we ever differentiate that? What sits under the very first square root sign? x plus y. Yes, exactly! After the first x, the rest is the same never-ending expression, and that is y itself. So y equals the square root of x plus y, and squaring, y squared equals x plus y. The forever has gone! Right? Now differentiate implicitly: two y y prime equals one plus y prime. Collect the y prime terms: y prime times two y minus one equals one. So y prime equals one over two y minus one. The double arrow means implies. Very good!

5:22Example 1: dy/dx for x2 + y2 = 25 and the slope at (3, 4)

Now, remember our circle from the start? Let's finish it. x squared plus y squared equals twenty five. Find d y by d x, and its value at the point three, four. Pause the video, and try it first. Done? Okay, let's check. Differentiate both sides with respect to x. x squared gives two x. y squared gives two y times y prime, not just two y, remember? And twenty five is a constant, so its derivative is zero. Okay? So y prime equals minus two x over two y, which is minus x over y, wherever y is not zero. See, no plus or minus, no square root. One clean formula. With me? Now put in the point three, four. What is the slope? Minus three over four. Correct! That is the slope of the tangent at three, four, and we never made y alone. Puzzle solved! Perfect!

6:28Example 2: dy/dx of the cycloid, using half angles

Now a harder one, also from the course. x equals a times t minus sin t, and y equals a times one minus cos t. This curve is called a cycloid. Find d y by d x. Pause and try. Hint: at the end, think of half angles. Done? Okay, let's check. Differentiate each with respect to t. d x by d t is a times one minus cos t. And d y by d t: the derivative of minus cos t is plus sin t, so it is a times sin t. Right? Divide, with the y part on top: a times sin t, over a times one minus cos t. It is correct, but the options may show a simpler form. So let's simplify. Okay? Now think. Which identities can we use on sin t and on one minus cos t? The half angle formulas, yes! Sin t equals two sin of t over two, times cos of t over two. And one minus cos t equals two sin squared of t over two. Clear? Put them in. The a on top and the a below cancel, the twos cancel, and one sin of t over two cancels. We are left with cos of t over two, over sin of t over two. That is cot of t over two. Beautiful, isn't it?

7:54Quick revision: implicit, parametric, never-ending roots

So, quick revision. Implicit: differentiate both sides, and y squared gives two y y prime, not two y. Parametric: d y by d x is d y by d t, over d x by d t, with y on top. And for a never-ending expression, replace the repeating tail by y. Now go and try the practice problems on this page. You can do them. All the best!