Find the derivative the following ways:
Using the Produc...

Question

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

g(t) = (t + 1)(t² - t + 1)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of H X equals 3 x + 2 multiplied by X2 + 4 X + 6 using the product rule. A says the derivative with 3 X2 + 22 X + 18. B 3 X2 + 24 X + 14, C 9 X2 + 26 X + 28, and the D 9 X2 + 28 X + 26. Now what do we know that can help us to find the derivative of H of X by the product rule? What does the product rule say? Well, basically, what it says is that if we have a function, let's say the function is H of X and it equals two terms being multiplied by each other, that is the product of U and V. Then the derivative of H X equals the derivative of u multiplied by V plus the derivative of v multiplied by u where U and V are functions of X. In other words, we can differentiate both of the terms and then multiply them by the opposite original function. So let's go ahead and do that, OK? In this problem, let's let U be 3 X + 2, and let's let V be X2 + 4 X + 6. Now if we let you equal 3 x + 2. Then we can differentiate you using the power rule. The derivative of 3 X would be 3, and the derivative of 2 would be 0 because 2 is a constant. On the other hand, if we let V be equal to X2 + 4 X + 6. Then the derivative of V equals 2 X, the derivative of X is 2 X. The derivative of 4 X is 4, and the derivative of 6 is 0. Know that we have U, V, and their derivatives, we can apply the product to it. Because no, that means then the derivative of H of X is going to be U3 multiplied by VX2 + 4 X + 6. Plus V, which is 2 X + 4 multiplied by U, which is 3 X + 2. Now we can go ahead and simplify. Now 3 multiplied by X2 + 4 X + 6 gives us 3 X2 + 12 X. OK. Plus 18. And no, if we were to solve 2 X + 4 multiplied by 3 X + 22 X multiplied by expression would be 6 X squared + 4 X, and 4 multiplied by the expression would be 12 X plus 8. Now if we look for alike terms, 3 x 2 and 6 X2d are alike, we can add them to get 9 X2, 12 X 4 X, and 12 X are alike, and when we add them, we get 28 X. and 18 and 8 are constants. 18 + 8 is 26. Therefore, the derivative of H of X is 9 X2 + 28 X + 26. If we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
g(t) = (t + 1)(t² - t + 1)

Key Concepts

Product Rule

Quotient Rule

Simplification of Derivatives

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