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

Question

f(x) = 3x³ + 5x² + 6x

Accepted Answer

Hello. In this video, we are going to be calculating the 1st, 2nd, and 3rd derivative of the given function. The function given to us in this problem is K X equal to 2x the power of 6 plus 4 X 5 plus 9 X cubed. How can we take the derivative of this type of function? Well, notice that we have the sum of multiple terms in the function, so we are going to have to take the independent derivatives of each term. Furthermore, each term is given to us in an exponential form. So we are going to need to take the derivatives of each term using the power rule for derivatives. Now, the power rule states the following. If we want to take the derivative of X to the power of N, we will move the value of the exponent in front of X, then decrease the value of the exponent by 1. So, how is that going to look? Well, let's go ahead and apply the power rule to the first term, to X to the power of 6. So K of X, which is the first derivative, will equal to the coefficient of 2 multiplied by the derivative of X to the power of 6. We will take the value of the exponent, which is 6, and move it in front of X. Then we will decrease the value of the exponent by 1, and that will give us the value of 5. Then we will repeat this process with the other two terms. The derivative of the second term will be the coefficient of 4 multiplied by the derivative of X to the power of 5. We will take the exponent and move it in front of X, which is 5, then decrease the value of exponent by 1, which will give us a value of 4. Then we will add the coefficient of 9 multiplied by the derivative of X cubed, which will be 3 X raised to the power of 2. Now we need to simplify. 6 multiplied by 2 will give us the value of 12, so the first term simplifies to 12 x to the power of 54 multiplied by 5 will give us 20, so the second term will simplify to be 20 multiplied by X to the power of 4, and 9 multiplied by 3 will give us 27, so we have 27 X squared. This is going to be the value of the first derivative by the power rule. Now let's go ahead and take the second derivative. In order to get the second derivative, we will take the derivative of the 1st derivative, K X, and again, we will apply the power rule to every term in K X. The second derivative will be labeled as K of X, and by using the power rule, the derivative will be defined as 12 multiplied by 5 X to the power of 4, plus 20 multiplied by 4 X cubed, plus 27. Multiplied by 2 X. Now we will simplify the coefficients. 12 multiplied by 5 will give us the value of 60, so we have 60 X raise the power of 4, 20 multiplied by 4 will give us 80, so we have 80 X cubed, and 27 multiplied by 2 will give us the value of 54, so we have 54 X, and this is going to be the second derivative of the original function. Now, we need the 3rd derivative, which will be labeled as K of X, and we will repeat this process by taking the derivative of the 2nd derivative of K X. That is going to be 60 multiplied by 4 X cubed plus 80 multiplied by 3 X squared. Plus 54 multiplied by 1, multiplied by X raised the power of 0. And further simplifying will give us K of X equal to 60 multiplied by 4, which is 240 X cubed, plus 80 multiplied by 3, which will be 240 X squared, plus 51 multiplied by 1, multiplied by X rates the power of 0. Anything raised to the power of zero is just going to be 1, and 1 multiplied by 1 is just 1, while 54 multiplied by 1 is just itself. So the final term will just be 54, and this is going to be the third derivative of the original function. So, I hope this video helps you in understanding how to take the 1st, 2nd, and 3 derivative of the given function, 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) = 3x³ + 5x² + 6x

Key Concepts

Differentiation

Power Rule

Higher-Order Derivatives

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