Second derivatives Find d²y/dx² for the following functions.

Question

y = √x²+2

Accepted Answer

Welcome back, everyone. Determine the second derivative for the function y equals square root of X cubed plus 21. For this problem we're going to rewrite the function from its radical form into exponential form, so we end up with Y equals x cubed plus 21 is the power of 1/2. And the reason why we do this is simply because we want to apply the power rule. So the first derivative. is going to be found by applying the chain rule, because we have an exponential function, we first will have to bring down the exponents, so this gives us one half, then we multiply that by the inner function, which is X cubed plus 21. And now it is going to be raised to the power of 1/2 minus 1 based on the power rule which gives us negative 1/2. But now the inner function is also a composite function, right? So we have to multiply by the derivative of X cubed plus 21. And now if we simplify, we're going to get a 1/2. In X cubed plus 21 it's the power of -12 multiplied by 3 x 2, right, because we are applying the power rule once again for X cubed and the derivative of 21 is 0. It is a constant. And now, of course, we can rewrite it as 3 x 2 divided by 2 multiplied by X cubed plus 21 to the power of -12. This is our first derivative. And now if we want to identify the second derivative, we'll have to apply the product rule. Of course we can apply the quotient rule if we want, but we can use this form, and we have to recall that if we define two functions, let's suppose that the first one is F is going to be 3 x 2 divided by 2. And our G, the second function within the product, is going to be X cubed plus 21 raise to the power of -12. What we want to do is find the derivative of each. So for F, we want to apply the power rule. So the derivative of X quad is 2 X and 2 X divided by 2 gives us X. So F is simply 3 X. And now for a G prime. Well, we have to apply the same rules as before. First of all, we apply the power rules, so we bring down -12. This gives us X cued plus 21. We decrease the exponent by 1, which gives us -3 halves, and then we multiply by the derivative of the inner function, and we already know that it is 3 x 2, right? So now if we simplify it, well, we're going to get negative 3 x 2 divided by 2 multiplied by x cubed plus 21 raise to the power of -3 halves, and now we can apply the product rule. Let's recall that this is FG plus FG. And then we have our F which is 3 X. We have to multiply that by G, which is XQ + 21 raise to the power of -15, and then we add FG. So our F is 3 x 2 divided by 2, and now our G, well, we are simply multiplying by negative. 3 x 2 divided by 2. Which is multiplied by X cubed plus 21 raises the power of -3 halves. So we essentially have our second derivative. What we want to do is just simplify it, and we are going to factor out execute plus 21 raise to the power of -3 halves. That's what we can do to begin with. So let's write x cubed plus 21, raise the power of -3 halves, and now let's consider the other terms. Well, we can also factor out 3 X, but let's not rush. Let's simply factor out XQ plus 21 raise to the power of -3 halves. And now if we look at the first term, well, we are going to have 3 X multiplied by X cubed plus 21, right, because if we multiply that by the outer factor, we're going to get -15 as the exponent in the first term. And now we are going to add 3 x 2 divided by 2 multiplied by 3 x 2 divided by 2, and we also have a negative sign. So let's add a negative sign, and this is going to give us 9 x to the power of 4th divided by 4. So now what we can notice is that we have that 4 in the denominator, right? So we can also factor it out. And what we're going to do is simply factor out the 4, right? So we're going to write it. In the denominator, which essentially means that we eliminate it from the second term in pregnancies, but we have to multiply the first term by 4, which gives us 3 multiplied by 4, and that's 12. So now let's keep simplifying. We can now rewrite our denominators. We have 4, and because we have a negative exponent, well, we can write X cubed plus 21 raise to the power of 3 halves in the denominator. And now we simply want to look at the numerators. So we get 12 x multiplied by X cubed, right? And this gives us 12 x to the power of 4th minus 9 X to the power of 4th, and essentially we get 3 X to the power of 4th. And now if we consider the other term, we have 12 Xs multiplied by 21. When we multiply those, we get 252 X. So we have successfully identified the second derivative and we have simplified it. Thank you for watching.

Second derivatives Find d²y/dx² for the following functions.
y = √x²+2

Key Concepts

Second Derivative

Chain Rule

Implicit Differentiation

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