Second Derivatives

In Exerci...

Question

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

y² = x² + 2x

Accepted Answer

Welcome back everyone. Using implicit differentiation, determine the first and the second derivative of the function below. Y2 equals 3x2 + 4 X. So what we're going to do for this problem is identify the first derivative to begin with. We have to find the derivative of each side of the given equation. So D divided by the X of the left hand side, which is Y2d, must be equal to D divided by the X of the right hand side, 3 x 2 + 4 X. Now on the left hand side, we have a composite function, so we have to apply the chain rule. Applying the power rule, we get 2 Y. And because Y is a function of X. According to the chain rule, we are multiplying by D Y divided by DX, which represents the derivative of Y with respect to X. On the right hand side, we're applying the power rule which gives us the derivative of 3 X2, that's 3 multiplied by 2 X. So we get 6 X plus the derivative of 4 X which is 4. We can divide both sides by 2. And essentially divided by y as well, since we are solving for D Y divided by D X. So we get 6 X + 4, the right hand side, divided by 2 Y. Now dividing both sides of the fraction by 2. We get 3 X. Plus 2 Divided by Y Well done, so we have the expression of the first derivative, and now we can identify the second derivative by performing implicit differentiation once again. This means that the second derivative of Y. With respect to X. Is going to be equal. 2 D divided by D X. Of 3 X + 2 divided by Y. And we can continue our equation. Essentially, what we're going to do is apply the quotient rule. It says that first of all, what we want to do is differentiate the top function. And multiply it by the bottom function, which gives us Y multiplied by D divided by DX of 3 X + 2. Then we are subtracting the top function, 3 X + 2. Which was multiplied by the derivative of the bottom function and that's d y divided by D X. All of that must be divided by the square of the bottom function, so we are dividing by y2d. And now we can evaluate each derivative. We're going to get Y multiplied by 3, the derivative of 3 X is 3, and the derivative of 2 is 0 because it's a constant. And then we are subtracting 3 X + 2 multiplied by Dy divided by DX and we have that from our previous answer. 3 X plus 2 divided by y. All of that is divided by y squared. So now what we're going to do. Is essentially find. The common denominator or The top expression within our numerator, right? So we have 3 Y. Minus 3 X + 2 squared divided by Y. So in this context, we can see that our common denominator is going to be Y. And we are dividing by y squared. So essentially, we can combine the two denominators, Y and Y squad, which gives us Y cubed and the denominator. And now considering the numerator, 3 Y is going to become 3y2d. And we are subtracting. 3 X + 2 squad. And that's it we have identified the 2nd derivative. You can write it down? That's our final answer to this problem. Well then, thank you for watching.

Second Derivatives

In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.

y² = x² + 2x

Key Concepts

Implicit Differentiation

First Derivative (dy/dx)

Second Derivative (d²y/dx²)

Watch next