Derivatives in Differential Form

Question

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Accepted Answer

Welcome back everyone. Find the Y for the simplicitly defined function X Y2 + 3 X to the power of 3/2 minus 2y equals 0. First of all, let's recall how we define differentials. D Y is equal to Y multiplied by D X. What we want to do is simply solve for Y and we can use implicit differentiation. So let's differentiate the left hand side. X Y 2 derivative plus 3 multiplied by the derivative of X the power of 3 halves minus 2 multiplied by y derivative should be equal to the derivative of 0 and the derivative of 0 which is 0, it's a constant, right? So now what we want to do is apply the power rule for X Y2. Let's recall that we're taking the derivative of X, our first function, multiplying by Y2 our second function. Adding the first function X multiplied by the derivative of the second function, so the derivative of Y2d. Then we're adding 3 multiplied by the derivative of X to the power of 3 halves. Here we can apply the power rule. We bring down 3 halves and the original exponent decreases by 1 unit. And then we're subtracting 2 multiplied by the derivative of Y. So now the derivative of X is 1, we got Y2d. Plus x multiplied by the derivative of Y22 Y multiplied by Y. So we are applying the power rule and the chain rule, because Y is a function of X. Then we're adding 9 halves. Acts to the power of one half. And subtracting toy. This is going to be equal to 0. What we're going to do is basically Leave the terms with Y on the left side of the equation. So we're going to get Y. In 2 X Y, we're factoring out. Minus 2 And this is going to be equal to. Negative of Y2. Plus 9 halves X is the power of 1/2 or square root of X. So we're basically moving the positive terms to the right, so they become negative. Now, what we can do is basically solve for Y. Y is going to be equal to the right hand side divided by the factor on the left hand side. We get negative. Y squared plus. 9 halves square root of X divided by we can factor out 2. X Y minus 1. Now, to avoid having a negative sign in front, we can just incorporate it into the factor within the denominator, which gives us Y2 + 9 1/2 square root of x. Divided by two imprecs, 1 minus xy. So we're flipping the signs of those terms within our factor. And now because we have a fractional term in the numerator, we can just find the common denominator, which is 2. And combine our denominators. So we get Y2 multiplied by 2. Divided by 2, which gives us the original term Y2 + 9 square root of X, which is also divided by 2. So now we have a common denominator, and everything is divided by 2 multiplied by 1 minus XY. From here, because we have a double division, we can write 2 Y 2 plus 9 square root of X in the numerator. And multiply 2 by 2 in the denominator to give us 4 multiplied by 1 minus XY. Now that we have our derivative, we can write. The DY expression, which is our derivative, toy squad. Plus 9 square root of X. Divided by 4 multiplied by 1 minus Xy multiplied by the X. And that will be our final answer to this problem. Thank you for watching.

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Key Concepts

Implicit Differentiation

Chain Rule

Differential Notation

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