Derivatives in Differential Form

Question

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

Accepted Answer

Welcome back everyone. Determine the D Y for Y equals 3 square root of X divided by 2 + square root of X. Let's recall that we can express the differential Dy as the derivative of a function Y multiplied by DX. So what we want to do for this problem is identify the derivative of Y. And because we have a Rational expression. This means that we have to apply the quotient rule. It says that first rule we're taking the derivative of the numerator. We squirreled off eggs. Multiplying it by the bottom function, which is 2 + square root of X. And then we are subtracting the top function, 3 square root of X, multiplied by the derivative of the bottom function. So the derivative of 2 + square root of X, all of that is divided by 2 + square root of X2, which is the square of the bottom function. And now for simplicity, we're just going to remember that the derivative of square root of X can be calculated as the derivative of X to the power of 1/2. Applying the power rule, we get 1/2 x to the power of -12, and that simplifies to 1 divided by 2 square root of X. So now, continuing with the quotient rule, we can then show that the derivative of 3 square root of X is 3 multiplied by 1 divided by 2 square root of X. So that's 3 divided by 2 square root of X. This is multiplied by 2 + square root of X. And then we are subtracting. We were root of eggs. Multiplied by the derivative of 2 is 0, so we are simply multiplying by the derivative of square root of X. Which is 1 divided by 2 square roots of X. All of that is divided by. 2 + square root of X2. Now what we're going to do is simply multiply both sides of our equation by square root of X to combine those denominators. This gives us 2 + square root of X2, we can also add 2 in the denominator, right? So let's multiply both sides by 2 square root of X. Meaning our denominator becomes 2 square root of X multiplied by 2 + square root of X2. And now considering the resultant numerator, we are going to have 3 multiplied by 2 plus square root of X. Minus We Square root of X. Multiplied by 1, right, because we are essentially multiplying by 1 divided by 2 square roots of X. This was our original expression, but now we have already multiplied both sides of the whole fraction by 2 square root of X. And now what we're going to do is distribute the terms, so we got 6. Plus 3 square root of X minus 3 square root of X divided by 2 square roots of X multiplied by 2. Less square root of X squared. We can cancel out the radical terms and the numerator. And then 6 divided by 2 gives us 3. So we end up with 3 in the numerator. And the denominator becomes square root of x multiplied by. 2 + square root of X2. Now returning to the differential expression, we can show that the Y equals. 3 divided by square root of X multiplied by 2 plus square root of x 2. And then multiplied by D X. That would be our final answer to this problem. Thank you for watching.

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

Key Concepts

Derivatives

Differential Form

Quotient Rule

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