Computer Explorations

Use a ...

Question

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

x√(1 + 2y) + y = x², P(1,0)

Accepted Answer

Hello. In this video, we are going to be finding the derivative DYDX for the given equation and evaluate the derivative at the point 2.0. So the equation given to us is X multiplied by the square root of 4 + 3Y + 2 Y equal to 2 X2 minus 4. Before we take the derivative, let's go ahead and rewrite the function as an exponential form, so we can rewrite the function as X multiplied by the quantity 4 + 3Y raised to the half power plus 2y equal to 2 X2 minus 4. Now, the problem wants us to find the derivative DYDX. In order to do this, we are going to take the derivative of the entire equation with respect to X. Now, for the first term, we have a product of a function of X and a function of Y. So in order to take the derivative of the first term, we will go ahead and apply the product rule. That is going to be the original first function multiplied by the derivative of the second function, 4 + 3Y raised to the half power. The derivative by the chain rule will be 1/2 multiplied by 4 + 3Y, multiplied by the derivative of the inside function. Which is going to be 3 multiplied by 1, but because this is a derivative of Y with respect to X, we will end up with 3 multiplied by DYD X. Then we will add the original second function, which is going to be 4 + 3Y raise to the half power, multiplied by the derivative of X with respect to X, which will be 1. Then we will take the derivative of 2 Y, which is going to be 2DYDX, and this will equal to the derivative of 2 X squad, which will be 4 X minus the derivative of 4, which is 0, since 4 is a constant. Let's go ahead and simplify this first term now. In order to make this exponent positive, we will go ahead and move this exponent to the denominator of 1/2. Then we will go ahead and multiply 3 and X together, and we can rewrite the first term to be 3 X divided by 2 multiplied by the square root of 4 + 3 Y, multiplied by DYDX. Plus the square root of 4 + 3 Y plus 2 multiplied by DYDX, and this will equal to 4 X. What we want to do in order to solve for DYDX is we want to factor out a DYDX from the left-hand side of the equation. But in order to do this, we will need to move all the terms without a DYDX to the right-hand side of the equation. We will go ahead and subtract the square root of 4 + 3Y to both sides of the equation, and that will leave us with 3 X divided by 2 multiplied by the square root of 4 + 3 Y. DYDX plus 2 multiplied by DYDX is equal to 4 X minus the square root of 4 + 3Y. Now what we are going to do is we are going to factor out the DYDX, and that is going to leave us with DYDX multiplied by the quantity, 3 X divided by 2 multiplied by the square root of 4 + 3 Y plus 2, and this is equal to 4 X minus the square root of 4 + 3Y. Now, in order to isolate DYDX, we will go ahead and divide. By the entire left hand quantity, and that is going to leave us with DYDX equal to 4 X minus the square root of 4 + 3Y, and this will be divided by 3X divided by 2 multiplied by the square root of 4 + 3Y. Plus 2 This is going to be the first part of our solution. This is the derivative DYDX. Now, for the second part of the problem, we want to evaluate this derivative at the point 2.0. So what we are going to do is we are going to plug in 2 for every X value, and 0 for every Y value within our derivative. That is going to give us 4 multiplied by 2 minus the square root of 4 + 3 multiplied by 0, which is 0. Divided by 3 multiplied by 2, divided by 2 multiplied by the square root of 4 plus 3 multiplied by 0, which is 0, and then plus 2. By simplifying the numerator, 4 multiplied by 2 will give us 8, and the square root of 4 will give us 2. 3 multiplied by 2 will give us 6. And the square root of 4 is 2, and 2 multiplied by 2 is 4. Then we will add 2 to this, but let's go ahead and rewrite 2 with a denominator of 4. So we can rewrite 2 to be 8 divided by 4. By combining the fractions in the denominator, we will have a denominator of 14. Yeah, 14 divided by 4, and 8 minus 2 will give us the value of 6. Finally, if we multiply by the reciprocal, we will end up having 6 multiplied by 4 divided by 14, which is going to give us 24 divided by 14, which will simplify to be 12. Divided by 7. And this is going to be the solution for the derivative evaluated at the point 2.0. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.

x√(1 + 2y) + y = x², P(1,0)

Key Concepts

Implicit Differentiation

Chain Rule

Evaluating Derivatives at a Point

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