Questions: Find the derivative the following ways. a. Using the Quotient Rule. b. By simplifying the quotient first. Verify that your answer agrees with part (a). h(w) = (2w^3 - w) / w a. Use the Quotient Rule to find the derivative of the function. Select the correct choice and fill in the answer boxes to complete your choice. A. The derivative is (w( ) + (2w^3 - w)( )) / (2w^3 - w)^2 B. The derivative is (w( ) - (2w^3 - w)( )) / 2w C. The derivative is (w( ) - (2w^3 - w)( )) / (2w^b - w)^2 D. The derivative is (w( ) + (2w^3 - w)( )) / w^2 E. The derivative is (w( ) - (2w^3 - w)( ) / w^2

Solution Steps

The given function is:

\[ h(w) = \frac{2w^3 - w}{w} \]

The Quotient Rule states that if you have a function \( h(x) = \frac{f(x)}{g(x)} \), then the derivative \( h'(x) \) is given by:

\[ h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} \]

For our function, \( f(w) = 2w^3 - w \) and \( g(w) = w \).

First, find the derivative of \( f(w) \):

\[ f'(w) = \frac{d}{dw}(2w^3 - w) = 6w^2 - 1 \]

Next, find the derivative of \( g(w) \):

\[ g'(w) = \frac{d}{dw}(w) = 1 \]

Substitute \( f(w) \), \( f'(w) \), \( g(w) \), and \( g'(w) \) into the Quotient Rule formula:

\[ h'(w) = \frac{w(6w^2 - 1) - (2w^3 - w)(1)}{w^2} \]

Simplify the expression:

\[ h'(w) = \frac{6w^3 - w - 2w^3 + w}{w^2} \]

Combine like terms:

\[ h'(w) = \frac{4w^3}{w^2} \]

Simplify further:

\[ h'(w) = 4w \]

Simplify the original function \( h(w) \):

\[ h(w) = \frac{2w^3 - w}{w} = 2w^2 - 1 \]

Now, find the derivative of the simplified function:

\[ h'(w) = \frac{d}{dw}(2w^2 - 1) = 4w \]

Both methods yield the same derivative:

\[ h'(w) = 4w \]

Final Answer