Derivatives Find the derivative of the following functions. Se...

Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(x) = 6x⁵ - 5/2 x² + x + 5

Accepted Answer

Hello. In this video, we are going to be finding the derivative of the given function. The function given to us in this problem is F of X equal to 7 X 6 minus 7/4 multiplied by X 4 + 3 X + 10. In order to take the derivative of the given function, we are going to have to take the derivative of the independent terms within F of X. Now, notice that the 1st 3 terms are in the form of an exponential value of X. In order to take the derivative, we can use the power rule for the derivatives. Let's say that we have a term of X in the form of X to the power of N. The power rule states that if we have a function of X in an exponential form, we can move the value of the exponent N in front of the value of X, then decrease the value of N by 1. So, let's go ahead and apply this to the 1st 3 terms of the given function. We will label the derivative as F of X. Then, for the first term, the derivative of 7 X to the power of 6 will be the coefficient of 7 multiplied by the derivative of X to the power of 6. In this case here, our N is going to equal to 6. So by the power rule, we will move the value of 6 in front of the variable X. Then we will decrease the value of our exponent by 1.6 minus 1 is going to be 5, and this will be the derivative of X to the power of 6. Then we will repeat this process for the 2nd term. We have negative 7/4 multiplied by the derivative of X to the power of 4. By the power rule, our end value is going to be 4. That is going to be 4 multiplied by X, raised to the power of 4 minus 1, which is 3. Then we will apply the power rule to 3X. Now, one thing to clarify is that the variable X has an exponent of 1, so our N value in this case is going to be 1. So the derivative of 3 X is going to be the coefficient of 3, multiplied by 1, multiplied by X, raising the power of 1 minus 1, which is 0. Now, what about the derivative of the constant value of 10? Well, if we are trying to take the derivative of a constant value, the constant rule states that the derivative of any constant is zero. That means that the derivative of 10 is going to be 0. Now what we need to do is we need to simplify the terms in our derivative. So, for the first term, 7, multiplied by 6 is going to give us the value of 42. So we have 42 X to the 5th power. Next, 7/4s multiplied by 4 is going to give us 7, so we have -7 multiplied by X cubed. Now, X to the power of 0 is going to simplify to be 1, because anything to the power of zero is just 1. 3 multiplied by 1, multiplied by 1, is just going to give us the value of 3. And this is going to be the simplest form of our derivative. F of X is equal to 42 X 5 minus 7 X cubed plus 3. And with that being said, the solution to this problem is going to be B. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.g(x) = 6x⁵ - 5/2 x² + x + 5

Watch next

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
g(x) = 6x⁵ - 5/2 x² + x + 5