Find the following higher-order derivatives.

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Question

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

Accepted Answer

Welcome back, everyone. Find the second derivative of the function F of X equals LN 3x2 + 2. So to find the second derivative, we first will want to identify the first derivative of the given function. So let's evaluate F of X, and this means that we want to evaluate the derivative of the natural log of 3 x2 + 2. And for this purpose we want to recall that l and of X derivative is equal to 1 divided by X. But in this case, well, we have a composite function, so first of all, we take 1 divided by 3 x 2 + 2 because this is what we have instead of X, and then we multiply by the derivative of 3 x2 + 2 based on the chain rule because this is our inner function. Which can be evaluated using the power rule. So we get 3 X2 derivative, which is 6 X, the derivative of 20, it is a constant. And then we are rewriting our denominator 3 X2 + 2. And this is our first derivative, and for the second derivative, we can either apply a quotient rule or we can apply the product rule. So essentially what we're going to do is apply the product rule because generally it gives us the simplified form of the final answer. So we're going to rewrite the first derivative S 6X multiplied by 3x2 + 2, raise the power of negative first. And in this case, we want to evaluate the second derivative. We can factor out the constant, and then we are evaluating the derivative of the product X multiplied by 3 X2 + to raise the power of negative first. Now what we're going to do from here is simply set G equals x. This is our first function in the product, and H equals 3x2 + 2 raises the power of negative 1. And according to the product rule, if we want to differentiate G multiplied by H, this gives us G H plus G H. So we need two derivatives. We need the derivative of G, which is 1, because that's the derivative of X and the derivative of H. According to the power rule, we bring down -1, and this gives us -3X2 + 2 raise the power of -1 minus 1, which is -2. And then we have to apply the chain rules so we multiply by the derivative of the inner function, which is 3x2 + 2. And we already know what it is, right? So we can say that this is overall negative. We X2 + 2 raise the power of negative. Second, multiplied by 6 X. So we can use those in finding the second derivative. So the second derivative is going to be equal to. 6, in C's, we want to take G which is 1, multiplied by H, so we're going to write 3 X2 + 2. Was the power of negative first, plus GH derivative, so G is X. And each derivative is 6 X. Multiplied by 3 x2 + 2a the power of negative. That would be our expression, and now we are nearly done because what we want to do is simplify. So the second derivative is equal to 6 inis we can rewrite. 3 X2 + 2 raised the power of negative first as 1 divided by 3 X2 + 2. And for the second term while we are subtracting. 6 X squad. Divided by We X2 + 2 squad. From here, what we're going to do is simply find the common denominator. So the second derivative. is equal to, we have 6, and then in Cs to find the common denominator. Well, we want to multiply and divide the first part by 3 X2 + 2. So we're going to get 3 X2 + 2. Minus X2. And at the bottom, the common denominator is going to be 3 X2 + 2 squad. Now we don't need to include rencies anymore, we simply have a product, and we can continue. So 3 X squared. Minus 6 X2 gives us -3 X2, right? So we can say that we end up with 6 multiplied by. 2 minus 3 x 2 divided by We X2 + 2 squad. Which we can rewrite as -6. In 3 x 2 minus 2, so we are factoring out the negative sign, right? So -3 X2 becomes 3 X2 and 2 becomes negative 2. Divided by 3 X2 + 22, and we can write the negative sign in front of the fraction. So that'd be our second derivative and thank you for watching.

Find the following higher-order derivatives.

d²/dx² (In(x² + 1))

Key Concepts

Higher-Order Derivatives

Chain Rule

Natural Logarithm Function

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