(2) $$ \frac{9 x-4 y}{3 x^{2}+2 y^{\frac{1}{2}}} $$

Question

UpStudy AI · Accepted Answer

You’ve provided the expression:

$$
\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}
$$

To assist you effectively, I'll explore a common operation performed on such expressions: **finding the derivative with respect to $$ x $$**. If you had a different operation in mind (like simplifying, evaluating, etc.), please let me know!

Assuming $$ y $$ is a function of $$ x $$ (i.e., $$ y = y(x) $$), we'll use the **quotient rule** for differentiation. The quotient rule states that if you have a function $$ \frac{u(x)}{v(x)} $$, its derivative is:

$$
\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^{2}}
$$

Let's apply this to your expression.

### Step 1: Identify $$ u $$ and $$ v $$

$$
u = 9x - 4y
$$
$$
v = 3x^{2} + 2\sqrt{y}
$$

### Step 2: Find $$ u' $$ and $$ v' $$

Differentiate $$ u $$ and $$ v $$ with respect to $$ x $$:

$$
u' = \frac{d}{dx}(9x - 4y) = 9 - 4\frac{dy}{dx}
$$

$$
v' = \frac{d}{dx}\left(3x^{2} + 2\sqrt{y}\right) = 6x + 2 \cdot \frac{1}{2}y^{-\frac{1}{2}} \cdot \frac{dy}{dx} = 6x + y^{-\frac{1}{2}} \frac{dy}{dx}
$$

### Step 3: Apply the Quotient Rule

$$
\frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{(9 - 4\frac{dy}{dx})(3x^{2} + 2\sqrt{y}) - (9x - 4y)(6x + y^{-\frac{1}{2}} \frac{dy}{dx})}{(3x^{2} + 2\sqrt{y})^{2}}
$$

### Step 4: Simplify (If Possible)

This derivative can be left in this form, but depending on the relationship between $$ y $$ and $$ x $$, further simplification might be possible. If $$ y $$ is explicitly defined in terms of $$ x $$, you can substitute that relationship to simplify the expression further.

### Example: If $$ y $$ is Constant

If $$ y $$ is a constant (i.e., doesn’t change with $$ x $$), then $$ \frac{dy}{dx} = 0 $$. Substitute this into the derivative:

$$
u' = 9
$$
$$
v' = 6x
$$

So the derivative simplifies to:

$$
\frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{9(3x^{2} + 2\sqrt{y}) - (9x - 4y)(6x)}{(3x^{2} + 2\sqrt{y})^{2}}
$$

Simplify the numerator:

$$
27x^{2} + 18\sqrt{y} - 54x^{2} + 24xy = -27x^{2} + 24xy + 18\sqrt{y}
$$

So,

$$
\frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{-27x^{2} + 24xy + 18\sqrt{y}}{(3x^{2} + 2\sqrt{y})^{2}}
$$

### Conclusion

Without additional context or specific instructions, I've demonstrated how to differentiate the given expression with respect to $$ x $$. If you intended a different operation or have more details about $$ y $$'s relationship with $$ x $$, feel free to provide them!
To find the derivative of $$ \frac{9x - 4y}{3x^{2} + 2\sqrt{y}} $$ with respect to $$ x $$, use the quotient rule: $$ \frac{d}{dx}\left(\frac{9x - 4y}{3x^{2} + 2\sqrt{y}}\right) = \frac{(9 - 4\frac{dy}{dx})(3x^{2} + 2\sqrt{y}) - (9x - 4y)(6x + y^{-\frac{1}{2}} \frac{dy}{dx})}{(3x^{2} + 2\sqrt{y})^{2}} $$ If $$ y $$ is constant, the derivative simplifies to: $$ \frac{-27x^{2} + 24xy + 18\sqrt{y}}{(3x^{2} + 2\sqrt{y})^{2}} $$

(2) \( \frac{9 x-4 y}{3 x^{2}+2 y^{\frac{1}{2}}} \)

Solución de tutoría real

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions