Find `(dy)/(dx)` if `y = 5x^(4) + 6x^(3//2) + 9x`.

Options

A. `20 x^(3) + 9x^(3//2) + 9`
B. `20 x^(3) + 9x^(1//2)`
C. `20x^(3) + 9x^(3//2)`
D. `20x^(3) + 9x^(1//2) + 9`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = 5x^4 + 6x^{3/2} + 9x\), we will differentiate each term with respect to \(x\).

### Step-by-Step Solution:

1. **Differentiate the first term**:
\[ y_1 = 5x^4 \]
Using the power rule, \(\frac{d}{dx}(x^n) = nx^{n-1}\), we get:
\[ \frac{dy_1}{dx} = 5 \cdot 4x^{4-1} = 20x^3 \]

2. **Differentiate the second term**:
\[ y_2 = 6x^{3/2} \]
Again applying the power rule:
\[ \frac{dy_2}{dx} = 6 \cdot \frac{3}{2}x^{(3/2)-1} = 6 \cdot \frac{3}{2}x^{1/2} = 9x^{1/2} \]

3. **Differentiate the third term**:
\[ y_3 = 9x \]
The derivative of \(x\) is 1, so:
\[ \frac{dy_3}{dx} = 9 \cdot 1 = 9 \]

4. **Combine the derivatives**:
Now, we can combine the derivatives of all three terms to find \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{dy_1}{dx} + \frac{dy_2}{dx} + \frac{dy_3}{dx} = 20x^3 + 9x^{1/2} + 9 \]

Thus, the final result is:
\[ \frac{dy}{dx} = 20x^3 + 9x^{1/2} + 9 \]