Questions: Use algebraic techniques to rewrite g(x) = (-5x^(9/4) + 3x^(13/4) + 2) / (x^(1/4)) as a sum of three terms. Then find g'(x).

Transcript text: Use algebraic techniques to rewrite $g(x)=\frac{-5 x^{\frac{9}{4}}+3 x^{\frac{13}{4}}+2}{\sqrt[4]{x}}$ as a sum of three terms. Then find $g^{\prime}(x)$.

Solution

Solution Steps

To rewrite the function $ g(x) = \frac{-5 x^{\frac{9}{4}} + 3 x^{\frac{13}{4}} + 2}{\sqrt[4]{x}} $ as a sum of three terms, we can divide each term in the numerator by the denominator $ \sqrt[4]{x} = x^{\frac{1}{4}} $. This will simplify the expression into a sum of terms with powers of $ x $. After rewriting, we can find the derivative $ g'(x) $ by applying the power rule to each term.

Step 1: Rewrite the Function

We start with the function

\[ g(x) = \frac{-5 x^{\frac{9}{4}} + 3 x^{\frac{13}{4}} + 2}{\sqrt[4]{x}} = \frac{-5 x^{\frac{9}{4}} + 3 x^{\frac{13}{4}} + 2}{x^{\frac{1}{4}}} \]

By dividing each term in the numerator by $ x^{\frac{1}{4}} $, we can rewrite $ g(x) $ as:

\[ g(x) = -5 x^{\frac{9}{4} - \frac{1}{4}} + 3 x^{\frac{13}{4} - \frac{1}{4}} + 2 x^{-\frac{1}{4}} = -5 x^{2.25} + 3 x^{3.25} + 2 x^{-0.25} \]

Step 2: Find the Derivative

Next, we find the derivative $ g'(x) $ using the power rule. The derivative of each term is calculated as follows:

For $ -5 x^{2.25} $, the derivative is $ -5 \cdot 2.25 x^{2.25 - 1} = -11.25 x^{1.25} $.
For $ 3 x^{3.25} $, the derivative is $ 3 \cdot 3.25 x^{3.25 - 1} = 9.75 x^{2.25} $.
For $ 2 x^{-0.25} $, the derivative is $ 2 \cdot (-0.25) x^{-0.25 - 1} = -0.5 x^{-1.25} $.

Combining these results, we have:

\[ g'(x) = -11.25 x^{1.25} + 9.75 x^{2.25} - 0.5 x^{-1.25} \]