Find the slope of the curve x²+y³=2 at each point where y=1 (s...

Question

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure).

Accepted Answer

Welcome back, everyone. For the point Y equals 1, calculate the slope of the curve X2 + Y 5th equals 10. Here we have a graph with our curve. Now, if we're going to calculate the slope of the curve at our point, first we'll need to find the derivative of our equation. And for this equation we can differentiate it implicitly. Now differentiating implicitly. With respect to X, yeah. We can find the derivative and then substitute the X and Y values for our point where Y equals 1, OK? So to differentiate that means no, we're gonna be finding the derivative of X squared with respect to X, the derivative of Y to the 5th with respect to X. And the derivative of tin. With respect to X. Now what are they going to be? Well, to differentiate X grade, we can use the pore rule of differentiation, and that leaves us with 2 X. To differentiate Y to the 5th, we can use the chain rule. We're going to bring down the power 5 multiplied by y to the 4th, multiplied by the derivative of Y with respect to X, treating Y as the inner function. And finally, the derivative of 10 with respect to X is 0 because 10 is a constant. Now, we can go ahead and solve for DYD X by subtracting 2 X from both sides and dividing by 5Y to the 4th. And when we do that. We're going to get DUIDX to be equal to negative to X divided by 5Y to the 4th. Know that we have the derivative of our function, we can find the values where what we can find the slope for the points where Y E equals 1. Now, if we look at our graph here, we can tell that there are two points where Y E equals 1, OK? Y is equal to 1 at the 31, and at the, at the 31, sorry, this is 31, and at the 0.31 OK? So we can find the slope at both of those points because both of those points will give us the X and Y values that we need to plug into DUYDX, OK? So, starting with -31. OK, at 31, then our slope. Our slope OK, is going to be equal to -2 multiplied by -3 divided by 5 multiplied by 1 to the 4th. That is gonna give us 65, OK? So at -31 our slope is 6/5. At 3 negative, sorry, at 31. OK, at 31. Then again we substitute our values. The derivative is going to be -2 multiplied by 3, divided by 5 multiplied by 1 to the 4th, which would be -65. Therefore, the slope of the tangent line is 65 at -31 and 65 at 31. Thanks a lot for watching everyone. I hope this video helped.

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>

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