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Find $d y / d x$ using any method.$$3 x^{3 / 4}-2 y^{2 / 3}+7 y^{2}-3 x=9 x^{2}-5 y^{4}$$

$$\frac{y^{1 / 3}\left(216 x^{5 / 4}+36 x^{1 / 4}-27\right)}{x^{1 / 4}\left(240 y^{10 / 3}+168 y^{4 / 3}-16\right)}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 8

Implicit Differentiation

Derivatives

Campbell University

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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for this problem. We have been given an equation and asked to find the derivative. Now, this is an excellent opportunity to use implicit differentiation. This would be a very difficult equation to solve for why, So we can leave it all jumbled up with our exes and wise together and in use implicit differentiation. Now, the key to that is to remember that why is a function of X Even though we haven't explicitly solved for it, why isn't just another variable? It is a function of X. So when I take the derivative, for example, I have a Y to the fourth that will be looking at a little bit. When I take that derivative, I take the derivative as if why was a variable on a becomes for Why cubed? But why isn't just a variable? It's a function. So the chain rule says I then have to take the derivative of that function with respect to our variable X. So every time we take the derivative of why we're gonna have to tack on that d Y d X in order to finish the differentiation. So let's go back to our equation here. First of all, three x to the 3/4. Well, bring down that exponents. So that's 9/4 Ex. Subtract one. So it's extra the negative 1/4. Next to bring down that two thirds, it becomes four thirds. Why I subtract one So it's to the negative one. Third times D y d x There's that chain rule that we talked about next. Seven. Wife Square to bring down the two sets. 14. Why again? We need the D Y DX derivative of negative three. X is just negative. Three. Hey, nine X squared becomes 18 x and five y to the fourth. We bring down that four so it's minus 20 y cubed and once more the change will says I need d Y DX. Okay, from that was the calculus portion of this problem from this point forward, this is an algebra problem we want to solve for D. Y d. X. So every term with D Y d X will go toe one side of the equal sign everything else will go to the other. And then we saw for D Y d. X. So they already have to, uh, de y dx terms on the left hand side. Let's bring everything over to that side. So I have one term with d y DX that I need to bring from the right hand side. That becomes positive. 20. Why cute do y dx. I'm gonna copy down the two terms I already have over here and this. Actually, instead of having to the negative one third, I'm going to write it in the denominator toe. A positive one third power and I have 14. Why de y dx now, every term that is not containing a d y d X goes to the right hand side. And as you can see, I have two of those that I'm going to need to bring over. So I have 18 x minus. I'm sorry. Plus, through the three next plus three minus 9/4 X and again, I'm gonna rewrite this one just like I rewrote the y. I'm gonna say this is 9/4 times X to the 1/4. Okay? And let's factor out our d Y DX. That leaves us with 20 y cute minus 4/3 wide to the one third plus 14. Why? And then over here I have just the same thing we had before. Okay? Now, to make this a little bit easier on our next step, each side with what's in the parentheses on the left and the other three terms on the right, I want to put these with a common denominator. Make these in tow one fraction. So on the left hand side, my denominator will be three. Why? To the one third? That means I'm multiplying 20. That becomes 60 y. Why do the third? Why? To the one third becomes Why? To the 13th the four doesn't change. Because that's just a four and 14 times three gives me 42. And now we're gonna have why? To the four thirds over on the right hand side, my denominator is going to be four x to the 1/4. Okay. Well, what does that give me? 18 times four. That's gonna be 72 X. I knew. Um, exponents gonna be 5/4 three times four. That's 12 times X to the 1/4 and then nine already has that denominator. So it stays there. Okay, Get myself a little more space. Now. We solve de y dx. Well, I'm going to be dividing by a fraction here. When you divide by a fraction, it's like multiplying by the reciprocal. So I'm gonna multiply by the reciprocal of that fraction in my parentheses. So what that gives me when I multiply That puts the three wide to the one third on the top times the numerator of what's already on the right and on the denominator. I'll have what's already in the denominator on the right times, the numerator of the left. Okay, so that's my answer. Now we can simplify this just a little bit more. Um, since I already have some things factored out, I think I'm just gonna factor anything else that I can. My numerator every term in there has a factor of three. So I can factor out one mawr three, and that's going to leave me with 24 x to the 5/4 plus four x to the 1/4 minus three. The denominator every term of those parentheses has a factor of two. So I could factor out an extra two, and that leaves me with 30. Why? To the 13th minus two plus 21. Why to the four thirds. So that is my final answer. It's rather messy, but that is the just the derivative Off are given equation

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