Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).

Question

f(x) = 1/x

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says determine the 1st, 2nd, and 3rd derivative of the given function. And we're given F of X is equal to 5 divided by X. We're also given 4 possible choices as our answers. For choice A, we have F of X is equal to 5 divided by X2. F of X is equal to 10 divided by X cubed, and F of X is equal to -30 divided by X to the fourth. For choice B, we have F of X is equal to -5 divided by X2. F of X is equal to 10 divided by X cubed, and F of X is equal to 30 divided by X4. For choice C we have F of X is equal to -5 divided by X2. F of X is equal to minus 10 divided by X cubed, and F of X is equal to -30 divided by X to the fourth. And finally, for choice D we have FX is equal to minus 5 divided by X2, FX is equal to 10 divided by X cubed, and F of X is equal to minus 30 divided by X4. Now we need to determine the 1st, 2nd, and 3rd derivative of our function F of X, which is equal to 5 divided by X. And since we have a fraction here that we're gonna need to take a derivative of, we're going to need to use our quotient rule. So we need to recall our quotient rule, which is the derivative of the quantity of U divided by V. And that is going to be equal to V multiplied by U minus U multiplied by V and that whole quantity is divided by V2d. So we're gonna apply our quotient rule in order to calculate our derivatives. So the first derivative we need to find is our first derivative, so we'll be looking for F of X. And this is going to be equal to the derivative of 5 divided by X. So here we're going to apply our quotient rule. So when we take a derivative of 5 divided by X, we're going to get X, which is our denominator, multiplied by the derivative of our numerator. We take a derivative of 5, since it's constant, it's derivative is going to be equal to 0, so we have X multiplied by 0. Minus our numerator, which is 5, multiplied by the derivative of our denominator, they're taking the derivative of X with respect to X will give me 1. So I'll have 5 multiplied by 1. And this whole quantity is divided by our denominator squared, so we'll divide it by X2. And simplifying this expression, this is going to be equal to minus 5 divided by X2. And so this is our expression for our first derivative. Now, we need to calculate our second derivative F of X, and we'll do that by taking a derivative of F of X. So this is going to be equal to the derivative of minus 5 divided by X2. So again, we'll use our quotient rule. So we'll have our denominator, which is X2, multiplied by the derivative of our numerator, and here we're taking a derivative of -5, which is constant, which is going to be zero. Then we'll have minus our numerator, which is -5. Multiplied by the derivative of our denominator, and the derivative of X2. With respect to X is going to give me 2 X. And then this whole quantity is divided by our denominator squared. So when a square x 2, I get X rated to the 4th. So some find this expression, this is going to be equal to. 10X Divided by X-rays to the 4th, which is going to be equal to 10 divided by X cubed. And so this is the answer for our second derivative. And now we need to calculate our 3rd derivative. So we're going to be looking for F of X. So here we're going to be taking a derivative of F of X, which is equal to 10 divided by X cubed. So again, we're taking a derivative of fraction, so we'll have F of X equal to and again applying our quotient rule, we'll have our denominator, which is X cubed, multiplied by the derivative of our numerator, which we take a derivative of 10, we get 0. Then we'll have minus our numerator, which is 10, multiplied by the derivative of our denominator, and taking a derivative of execute will give us 3 X2. And then this whole quantity is divided by our denominator squared, and so when a square X cube, I'll get X rates to the 6th power. So simplifying this expression, this is going to be equal to minus 30, multiplied by X2, divided by X rates to the 6th. Which is going to be equal to -30 divided by X rates to the 4th. So this is our expression for our 3rd derivative. So now we have all of our derivatives calculated, we can compare our answers to our answer choices, and the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).
f(x) = 1/x

Key Concepts

First Derivative

Second Derivative