Find f′(x), f′′(x), and f′′′(x) for the following functions.

Question

f(x) = 3x² + 5e^x

Accepted Answer

Hello. In this video, we are going to be determining the 1st, 2nd, and 3rd derivative of the given function. The function given to us is G of X equal to 4 X cubed plus 7 multiplied by E raised to the power of X. So, how do we take the first derivative of this function? Well, since we have a sum of two terms, we are going to have to take the derivatives of the independent terms. So let's go ahead and start by calculating the derivative of 4 X cubed. Now, since we are given the term in the form of an exponential value, Let's go ahead and use the power rule for the first term. The power rule states the following. If we would like to take the derivative of X raised to the power of N, then the derivative is defined as the following. We will take the value of our exponent and move it in front of X. Then we shall decrease the value of the exponent by 1. So how is that derivative going to look? While the first derivative G of X will have the first term 4 multiplied by the derivative of X cubed. Now by the power rule, we will take the value of the exponent, which is 3, and move it in front of X. Then we will decrease the value of the exponent by 1, which will give us a value of the exponent by 2. So the derivative of X cubed will be 3 x 2. Now, what about 7E raise to the power of X? Well, since our variable X is in the exponential spot of Euler's constant E, we are going to have to use the exponential rule of the derivative. The exponential rule states the following. If we would like to take the derivative of Ease the power of X, we will first take the original function, which is E erase the power of X, multiply that by the natural logarithm of the base, which in this case is E, then multiply that by the derivative of the exponential function X. So, how is this derivative going to look? Well, that is going to be the coefficient of 7 multiplied by the derivative of Ease the power of X, and by the exponential rule that derivative is going to be E raise the power of X, multiplied by the natural logarithm of E, multiplied by the derivative of X. So let's go ahead and simplify. For the first term, 4, multiply by 3, will give us the value of 12, so we will have 12 X squared. Next, the natural logarithm of E will simplify to be 1. And by the power rule, the derivative of just X is going to be 1. That means that the derivative of the second term is going to be 7 E rate the power of X, and that means that the first derivative of the original function will be G equal to 12 X2 + 7 multiplied by E raise the power of X. Now we need to calculate the 2nd derivative. In order to calculate the second derivative, we will take the derivative of the 1st derivative. The second derivative will be labeled as G of X. We will use the power rule on the first term, which is going to be 12, multiplied by 2 X, and the exponential rule on the second term, which is going to follow a similar argument as the first derivative. It will be 7 multiplied by Ease the power of X, multiplied by the natural logarithm of E, multiplied by the derivative of just X. And we already know that this is just going to simplify to be erased the power of X. That means that the second derivative G of X will equal to 12 multiplied by 2, which is 24 X plus 7 E to the power of X. This is going to be the 2nd derivative. Now what about the 3rd derivative? Well, in order to calculate the 3rd derivative, we will just take the derivative of the 2nd derivative. The 3rd derivative will be labeled as G of X, and by the power rule that will be 24 multiplied by 1 multiplied by X rates to the power of 0, plus 7 multiplied by the derivative E to the power of X, which we already know is going to be E to the power of X. And that means that the triple derivative G of X is going to simplify to be 24 + 7 multiplied by E to the power of X. And that is going to be the 1st 3 derivatives of the original function. So, I hope this video helps you in understanding how to calculate the 1st 3 derivatives, and I will go ahead and see you all in the next video.

Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x² + 5e^x

Key Concepts

Differentiation

Higher-Order Derivatives

Exponential Functions

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