Second Derivatives

Find y'' ...

Question

Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

Accepted Answer

Determine the second derivative of Y equals 2 + 2 X, all risk the negative 1 power. Now, to find the 2nd derivative, we need to find the 1st derivative first. We will say Y prime. Now, this is actually a chain rule. Our outer derivative, you will find first. And then we'll find out. Our outer derivative is a power rule, where we bring down the -1 from the exponent, multiply it by 2 plus the square of X, and then decrease our exponent by 1. To -2. Now, this is also a chain rule. So we need a derivative of the inside function 2 plus the square root of X. We have 11 multiplied. By 2 plus the square of X to the negative. Which is all multiplied by the derivative of 2 + square root of X. Which is 1 divided by 2 square roots of X. Now, this is our first derivative. We can now find our 2nd derivative. Second derivative is given by a product rule. So, We have The derivative of -1 multiplied by 2 + square root of X to the negative, all multiplied by 1 divided by 2 square roots of X. We'll take the product rule and have -1 on the outside of everything. Multiplied By 2, multiplied by 2 plus 2 X. Race to the negative 3 power. Which is multiplied by the interior derivative, one divided by 2 square roots of X. This is all multiplied by 1 divided by 2 squares of X for a rightmost products. We will now add the other derivative. -1, multiplied by 2 plus the square x, rates negative, multiplied by the derivative of 1 divided by 2 square roots of X. This is 1/2 multiplied by X to negative 1/2, and that derivative. will be 1/4 X 3 halves. So we can multiply this. To get our product rule. Now we have to simplify everything. We have Negative 1. Divided by 2 X. Multiplied By 1/2. Which gives us -1. Divided by 4 X. All multiplied By -2 2 + square of X. To the negative 3 So we will bring this farther down. And we'll look at the right most products. But that doesn't simplify right now, so we'll just leave it. We will add Or subtract because it's negative. 2 plus the square of X, raised to the negative second, multiplied. By 14th. X The negative 3 halves. Now let's see if we're able to simplify a little bit more. We can multiply the negative 2 to get 1 divided by 2 X. Multiplied by 2 + square X to the negative3 power. Plus 14th multiplied. By 2 plus the square of X to the negative. Multiplied by X to the negative, 3 halves power. Now we can fix our negative exponents. We have 1 divided by 2 X multiplied. By one Divided by 2 + square of X, raise the 3. Plus 14th multiplied. By 1, divided by 2 plus the square of X squared. Multiplied by 1, divided by X to the 3 halves power. And try to keep simplifying. And we get one. Divided by 2 X multiplied by 2 + square x, rates the 3. Plus One Divided by 4 X, the 3 halves multiplied by 2 plus the square of X squared. We can keep simplifying. But if you get a common denominator, you should eventually end up with 2 + 3 squares of X. Divided by 4 X Square of X multiplied. By 2 plus a squad of X, cuts. And that should be your final. Simplification for your second derivative. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

Key Concepts

Chain Rule

Power Rule