Use implicit differentiation to find dy/dx.
e^xy

Question

Use implicit differentiation to find dy/dx.

e^xy = 2y

Accepted Answer

Welcome back everyone. Find the Dy divided by the X by implicit differentiation. We're given an equation E to the power of 3 XY equals 10 Y. If we want to find the first derivative, what we're going to do is simply set the derivative of the left hand side, which is the derivative of E to the power of 3 XY. And we're going to set this derivative equal to the derivative of the right hand side. So if we're given an equation, we can simply apply the first derivative to each side without rearrangement. From here, what we want to do is simply recall that the derivative of e to the power of X is e to the power of X based on the table of basic derivatives. So the derivative on the left becomes each to the power of 3 XY, but because the exponent is a function based on the chain rule, we have to multiply that by the derivative of 3 XY. And then on the right hand side we can factor out the constant which is 10 and multiply that by y because we're. Applying the derivative to Y. Now, what we notice is that we want to evaluate the derivative of X Y. Because we can factor out the constant, let's show that we're going to get 3 E to the power of 3 XY multiplied by the derivative of XY is equal to 10. And for this purpose we once applied the product rule. Let's say that x is our function F, even though it's an independent variable, and G is our function Y. So the product rule says that if we want to find the derivative of FG, this would be FG plus F G. So we want to identify F which is DX divided by DX and that's 1. And G prime is simply Y. So in this context, if we apply the product rule, then The derivative of XY is going to be equal to Y plus XY. So now substituting this into our expression, we get 3 E's, the power of 3 XY. Multiplied by y plus xy. Is going to be equal to 10. And from here our goal is to solve for a y prime. So to solve for Y, we want to distribute the terms first and we're going to get 3 Y. E to the power of 3 XY. Plus 3 X. Y eats the power of 3 XY equals 10 Y. And now we can label the terms that we want to combine, right, because those terms have Y. And this means that we want to move one of our terms to the right, or we can essentially say that. We can move 10 to the left and then move 3 Y each to the power of 3XY to the right. So it depends what's more preferred. For simplicity, we can just move 3 x ye to the power of 3 XY to the right. This will allow us to isolate the Y term and overall we're going to get 3 Y E to the power of 3 XY on the left. And on the right, we're going to get 10 Y minus because we are subtracting right, we're flipping the sign from positive to negative. 3 x y prime. Eats the power of 3 XY. From here, we're going to factor out the Y because we want to solve for it, so we get 3 YE to the power of 3 XY equals Y. In Cs, we're going to get 10 minus 3XE to the power of 3 XY. And now solving for Y, we are going to take the left hand side, which is 3 YE to the power of 3 XY. And divide that by the factor on the right hand side adjacent toy, which is 10 minus 3XE to the power of 3 XY. And we are done here, but we can also write the exponential term in front. We can just add a negative sign in front of the fraction and then flip the signs in the denominator. So our numerator remains the same. We have 3 y each the power of 3 x y and then we're flipping the signs within the denominator. So first of all we have positive. 3 x eats the power of 3 x y and the negative 10. Which is our final answer to this problem. Thank you for watching.

Use implicit differentiation to find dy/dx.
e^xy = 2y

Key Concepts

Implicit Differentiation

Chain Rule

Exponential Functions

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