Find the derivatives of the functions in Exercises 1–42.

Question

𝔂 = (x + 1)² (x² + 2x)

Accepted Answer

Welcome back everyone. Calculate the derivative of the function F of X equals X + 1, raise the power of 5 multiplied by X the power of 4 + 5 X. What we're going to do for this problem is simply apply the product rule for differentiation. We want to identify F of X, and according to the product rule we're taking the derivative of the first function. It is X + 1 raise the power of 5. So let's write it down. The derivative of X + 1 raise the power of 5. We're multiplying that by the second function. In this case it is X to the power of 4 + 5 X. And then we are adding the first function, X + 1 raises the power of 5, which is multiplied by the derivative of the second function X is the power of 4 + 5 x derivative. Now let's go ahead and evaluate each derivative. What we have to notice is that X + 1 raises the power of 5 is a composite function, right? The outer function contains an exponent. So this means that we are applying the power rule, and it says that we're bringing the exponent down so it becomes our constant 5, which is then multiplied by X + 1, the inner function raises the power of 4 because we are decreasing the exponent by 1 unit. And now according to the chain rule we are multiplying the result by the derivative of the inner function, which is the derivative of X + 1. Then we're going to multiply by the second function. Nothing changes, X is the power of 4 + 5 X. And then let's consider the second set of terms. So we're adding X + 1 raises to the power of 5. Which is multiplied by The derivative of X is the power of 4 + 5X, so we can differentiate each term using the power rule. The derivative of X is the power of 4 is 4 x cubed, and the derivative of 5 X is 5 because the derivative of X is 1, so we're simply adding 5. Now what is the derivative of X + 1? Well, it is equal to 1 because the derivative of X is 1 and the derivative of a constant is 0. So what we have to do is simplify. We're going to have 5 multiplied by X plus 1 raise the power of 4, multiplied by X to the power of 4 + 5 X. Plus X + 1 erase the power of 5. Multiplied by 4 x cubed plus 5. If we want to efficiently simplify the result, we're going to perform factorization, so we can factor out X + 1, raise the power of 4 because it is a repeating term, and then in paris, considering the first product, we will have 5. Multiplied by X to the power of 4 + 5 X. Well done. And now considering the second set, we have X + 1 because it was raised to the power of 5, right? Multiplied by 4 X cubed plus 5. So let's simplify. We have X plus 1, raise the power of 4. Let's distribute the terms, so we get 5 X the power of 4 plus 25 X. And now let's apply the foil property for X + 1 multiplied by 4 X cubed plus 5. This gives us 4 X to the power of 4 + 5 X plus 4 X cubed. Plus 5. All that we have to do is combine the like terms, which gives us X + 1 raise the power of 4. Multiplied by 9 X to the power of 4, because we have 5 X the power of 4 plus 4 X the power of 4. Plus 4 XQ. Right, we have 4 X cube. There are no more cubes, followed by. 25 X plus 5X, that's 30 X plus 5, which is our final answer to this problem. Thank you for watching.

Find the derivatives of the functions in Exercises 1–42.

𝔂 = (x + 1)² (x² + 2x)

Key Concepts

Derivative

Product Rule

Chain Rule

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