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Problem 99 Hard Difficulty

If $ = f(u) $ and $ u = g(x), $ where $ f $ and $ g $ are twice differentiable functions, show that
$ \frac {d^2 y}{dx^2} = \frac {d^2 y}{du^2} (\frac {du}{dx})^2 + \frac {dy}{du} \frac {d^2 u}{dx^2} $

Answer

$\frac{d^{2} y}{d u^{2}}\left(\frac{d u}{d x}\right)^{2}+\frac{d y}{d u} \frac{d^{2} u}{d x^{2}}$

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Video Transcript

all right, So we're given a formula and were asked to show that it's true and we know that Why is the outside function and you is the inside function? And so the first thing we should do is find D Y d. X. So that's going to be D y. Do you? That would be the derivative of the outside function times to utx the derivative of the inside function. And the convenient thing about this is, if you think of it sort of like fractions. It's almost like the D use Cancel on the top of the bottom, and what you see is D Y DX, so that's very convenient and helps us with the next steps. So now we want to find the second derivative D squared Y DX squared, and what we're going to do is use the product rule on what we just did. So we start with the first, do you? Why do you and multiply it by the derivative of the second on the derivative of D. U D X would be the second derivative of you d squared you d X squared. Now we add the second time's a derivative of the first. So the second is Dut X and the derivative of the first would be the second derivative of why. But remember that we have to use the chain rule every time we're differentiating Why? So we're going to have d squared. Why do you squared? And then times a derivative of the inside which is dut X So notice In our second term, we have dut x twice so we can go ahead and square that and we have d Why do you times d squared you d x squared? Plus do you d x squared times d squared? Why do you squared now That looks like the formula that were asked to prove except the order is a little bit different. So if we want to change the order, we end up with this term first and switch the order of those two just because we wanted to look identical to what they have in the book. So we have d squared. Why do you squared times, Do you d x quantity squared? Plus, now we're going with this second So plus d, why do you times d squared you d x squared