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Chapter 3: Problem 72

a. Calculate \(\frac{d}{d x}\left(x^{2}+x\right)^{2}\) using the Chain Rule. Simplify your answer. b. Expand \(\left(x^{2}+x\right)^{2}\) first and then calculate the derivative. Verify that your answer agrees with part (a).

Short Answer

Expert verified
Question: Find the derivative of the function \((x^2 + x)^2\) using (a) Chain Rule, and (b) by expanding the expression first. Verify that the answers are the same. Solution: (a) Using the Chain Rule, we found that: \(\frac{d}{dx}((x^2 + x)^2) = 2(x^2 + x)(2x + 1).\) (b) Expanding the expression first and finding the derivative, we have: \(\frac{d}{dx}(x^4 + 2x^3 + x^2) = 4x^3 + 6x^2 + 2x.\) After comparing both results and factoring the right side of part (b), we can see that they are the same: \(\frac{d}{dx}((x^2 + x)^2) = 2(x^2 + x)(2x + 1).\)

Step by step solution

01

Identify the outer and inner functions

We have the function \((x^2 + x)^2\). The outer function is \(u^2\) and the inner function is \(x^2 + x\). Let's call the inner function \(u\), so we have: \(u = x^2 + x\)
02

Calculate the derivative of the inner function

To apply the Chain Rule, we first need to find the derivative of the inner function \(u\). Apply the power rule and sum rule to find the derivative of \(u\): \(\frac{d}{dx}(u) = \frac{d}{dx}(x^2 + x) = 2x + 1.\)
03

Apply the Chain Rule

Now, we will apply the Chain Rule to differentiate the function \((x^2 + x)^2\). We have \(\frac{d}{dx}((x^2 + x)^2)= \frac{d}{dx}(u^2) * \frac{du}{dx} = (2u) * (2x + 1).\)
04

Replace u with the inner function

Now, replace \(u\) with the inner function \(x^2 + x\) to get the final derivative: \(\frac{d}{dx}((x^2 + x)^2) = 2(x^2 + x)(2x + 1).\) #b. Expanding the expression first#
05

Expand the expression

First, expand \((x^2 + x)^2\). We have: \((x^2 + x)^2 = (x^2 + x)(x^2 + x).\) Expanding this using the distributive property, we get: \((x^2 + x)(x^2 + x) = x^4 + 2x^3 + x^2.\)
06

Calculate the derivative of the expanded expression

Now, we differentiate the expanded expression \(x^4 + 2x^3 + x^2\). Calculate its derivative using the power rule and sum rule: \(\frac{d}{dx}(x^4 + 2x^3 + x^2) = 4x^3 + 6x^2 + 2x.\)
07

Verify that the answers from parts (a) and (b) are the same

Now, let's verify that the answers from parts (a) and (b) are equal. From part (a), we have: \(\frac{d}{dx}((x^2 + x)^2) = 2(x^2 + x)(2x + 1).\) From part (b), we have: \(\frac{d}{dx}(x^4 + 2x^3 + x^2) = 4x^3 + 6x^2 + 2x.\) Factor the right side of part (b) to check if it is equal to the derivative we found in part (a). Factoring, we obtain: \(2x(2x^2 + 3x + 1) = 2(x^2 + x)(2x + 1).\) Since both answers are equal, we can conclude that the derivatives found using the Chain Rule and by expanding the expression first are the same.

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