In Exercises 43–50, find by implicit differentiation.

Question

y² = x .

x + 1

Accepted Answer

Welcome back, everyone. By using implicit differentiation, find D Y divided by D X for the curve given by the equation Y2 equals 2 x minus 3 divided by X + 2. Let's begin by understanding that Y is a function of X, such that Y equals Y of X. If we want to use implicit differentiation, this means that we're taking the original equation. Differentiating the left hand side and setting it equal to the derivative of the right hand side. So our goal for this problem is to find the derivative of the left side and the right side. Let's begin with the left side. We want to identify the derivative of Y2d. Let's remember that it is a composite function. So first of all, we are differentiating Y2d with respect to Y, which is to Y, according to the power rule, but because Y is a function of X, we have to apply the chain rule and multiply by Y. Y is equal to the Y divided by DX. We're just going to use that shorthand notation. So we understand that on the left we are getting 2 Y multiplied by Y. For the right hand side we simply want to apply the quotient rule, so let's differentiate 2 x minus 3 divided by X + 2. According to the quotient rule, we are first of all taking the derivative of the numerator, so the derivative of 2 x minus 3, which is multiplied by the denominator X + 2, we are subtracting the numerator to x minus 3 multiplied by the derivative of the denominator X + 2 derivative, and all of that is divided by the square of the denominator, so X plus 2 squad. Now the derivative of 2 X is 2 based on the power rule, the derivative of -3 is 0, and the derivative of X + 2 is 1 because the derivative of X is 1 and the derivative of 2 is 0. So from here we can simplify the result and distribute the terms. We get 2 x + 4. Minus 2 x + 3 divided by x + 2 squad. We can cancel out 2 X and -2 X, which gives us 7. Divided by X + 22. And we're going to substitute this result into our equation. So on the right hand side, the overall derivative is 7 divided by X + 22. And now if we're solving for Y, we simply have to divide both sides by 2 Y, which gives us our function on the right hand side. Divided by 2 y so we simply can write 2 Y in the denominator. So Y equals 7 divided by 2y. Which is multiplied by x + 2 squad, and that's our final result which is D Y divided by D X. So we can label it as our final answer to this problem. Thank you for watching.

In Exercises 43–50, find by implicit differentiation.

y² = x .
x + 1

Key Concepts

Implicit Differentiation

Chain Rule

Derivative of a Function

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