Find and simplify the derivative of the following functions.

Question

h(x) = (5x⁷ + 5x)(6x³ + 3x² + 3)

Accepted Answer

Hi, everyone, let's take a look at this practice problem dealing with derivatives. This problem says, determine the derivative of the function PF X, which is equal to the quantity of 2 X 5 minus 4 X in quantity, multiplied by the quantity of 3X4th, minus 2x2 + 1. So this problem asks us to calculate the derivative of our function p of X. In order to take this derivative, we're going to need to use our product rule. So recall for your product rule that we have the derivative with respect to X. Of the quantity of FX. Multiplied by G of X. And this is going to be equal to F of X. Multiplied by G of X. Plus G of X. Multiplied by F of X. So here we're gonna use our product rule to calculate our derivative. So we're gonna be looking for P of X. And this is going to be equal to our first term, which is the quantity of 2 X to the 5th minus 4 X. In quantity, and that gets multiplied by the derivative with respect to X of the second term in our product, which is the quantity of 3 X to the 4th, plus 2 X squad + 1 in quantity, then we'll have plus the 2nd term in our product, which is the quantity of 3 X to the 4th. Plus 2 X squared. Plus 1 in quantity, multiplied by the derivative with respect to X of our first term in a product, which is the quantity of 2 X to the 5th minus 4 X. So now we need to just take the derivatives in our expression here. This is going to be equal to the quantity of 2 X 5. Minus or X. In quantity, multiplied by the quantity. Of here we take the derivative of 3 X to the fourth, and when I do that, I'll get 12 x cubed. Then I'll have plus the derivative of our second term here of 2 X squared, it's gonna give me 4 X. Then we'll take the derivative of 1 with respect to X, which is 0. We'll end our quantity after the 4 X term. Then we'll have plus the quantity of 3X4 plus 2 X squared. Plus 1 in quantity, multiplied by the quantity. I want to take a derivative of our first term of 2 X to the 5th, I'll get 10 X to the 4th. Then we'll have minus the derivative of 4 X, which is just 4. So now I just need to perform the multiplication, so this is going to be equal to. 24 X rated to the 8th. Plus 8 x 6. -484. Minus 16 X2. Plus 30 X 8. Plus 20 x 6. Plus 10 14th. Minus 12 x to the 4th. Minus 8 x 2. Minus 4. So now we need to just collect like terms. So this is going to be equal to. If I look at the terms that have X-rays to the 8th, I have 24 X to the 8th, plus 30 X to the 8th, and so that'll give me 54. Multiplied by X-rays to the 8th. Next, I'll look for terms that have X to the 6th in them, so I'll have 8 X rays to the 6th. And then I also have plus 20 X to the 6th, so that will give me plus 28 X to the 6th. Then if I look at X to the 4th terms, we'll have 48 X4. Less 10, 4th, minus 12 x 4th, and so that's going to give me minus 50 X to the 4th. Looking at the quadratic terms, we'll have -16 X2, minus 8 X2, which could be minus 24, X2, and then we'll have for a constant term. Just minus 4. So this here is the answer that we were actually looking for. So our final answer is P X is equal to 54 X 8 plus 28 x 6th, minus 50 X 4th, minus 24 X squad minus 4. So, I hope that this explanation has been useful, and I'll see you in the next video.

Find and simplify the derivative of the following functions.
h(x) = (5x⁷ + 5x)(6x³ + 3x² + 3)

Key Concepts

Derivative

Product Rule

Simplification of Derivatives

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