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Chapter 3: Problem 40

Find the derivative of the following functions. $$y=\frac{w^{4}+5 w^{2}+w}{w^{2}}$$

Short Answer

Expert verified
Answer: \(\frac{dy}{dw} = \frac{2w^5 - w^2}{w^4}\)

Step by step solution

01

Find the derivatives of f(w) and g(w)

First, we need to find the derivatives of \(f(w)\) and \(g(w)\). We can find the derivatives of these functions by applying the power rule, which states that when finding the derivative of a function of the form \(aw^n\), the derivative is given by \(anw^{n-1}\), where a and n are constants. For \(f(w) = w^4 + 5w^2 + w\), we have: $$f'(w) = \frac{d}{d w}(w^4) + \frac{d}{d w}(5w^2) + \frac{d}{d w}(w)$$ $$f'(w) = 4w^3 + 10w + 1$$ For \(g(w) = w^2\), we have: $$g'(w) = \frac{d}{d w}(w^2)$$ $$g'(w) = 2w$$
02

Apply the quotient rule

Now that we have the derivatives for \(f(w)\) and \(g(w)\), we can apply the quotient rule to find the derivative of the given function. Recall that the quotient rule formula is: $$\frac{d}{d w} \left(\frac{f(w)}{g(w)}\right) = \frac{f'(w)g(w) - g'(w)f(w)}{[g(w)]^2}$$ Applying the quotient rule, we have: $$\frac{d}{d w} \left(\frac{w^4 + 5w^2 + w}{w^2}\right) = \frac{(4w^3 + 10w + 1)w^2 - (2w)(w^4 + 5w^2 + w)}{(w^2)^2}$$
03

Simplify the expression

Next, we can simplify the expression by expanding and combining like terms: $$\frac{d}{d w} \left(\frac{w^4 + 5w^2 + w}{w^2}\right) = \frac{(4w^5 + 10w^3 + w^2) - (2w^5 + 10w^3 + 2w^2)}{w^4}$$ By combining like terms, we get: $$\frac{d}{d w} \left(\frac{w^4 + 5w^2 + w}{w^2}\right) = \frac{(4w^5 - 2w^5) + (10w^3 - 10w^3) + (w^2 - 2w^2)}{w^4}$$ Finally, we simplify the expression: $$\frac{d}{d w} \left(\frac{w^4 + 5w^2 + w}{w^2}\right) = \frac{2w^5 + 0 - w^2}{w^4}$$ Thus, the derivative of the given function is: $$\frac{dy}{dw} = \frac{2w^5 - w^2}{w^4}$$

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