2. Differentiate each of the following, (d). $$ \frac{3 x}{a-b x^{2}} $$

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To differentiate the function $$ f(x) = \frac{3x}{a - b x^{2}} $$ with respect to $$ x $$, we can use the **quotient rule**. The quotient rule states that if you have a function $$ f(x) = \frac{g(x)}{h(x)} $$, then its derivative is:

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

Let's apply this rule step by step.

### Step 1: Identify $$ g(x) $$ and $$ h(x) $$
$$
g(x) = 3x \
h(x) = a - b x^{2}
$$

### Step 2: Find $$ g'(x) $$ and $$ h'(x) $$
$$
g'(x) = \frac{d}{dx}(3x) = 3 \
h'(x) = \frac{d}{dx}(a - b x^{2}) = -2b x
$$

### Step 3: Apply the Quotient Rule
$$
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
$$
$$
f'(x) = \frac{3(a - b x^{2}) - 3x(-2b x)}{(a - b x^{2})^{2}}
$$

### Step 4: Simplify the Numerator
$$
3(a - b x^{2}) + 6b x^{2} = 3a - 3b x^{2} + 6b x^{2} = 3a + 3b x^{2}
$$

### Step 5: Write the Final Derivative
$$
f'(x) = \frac{3a + 3b x^{2}}{(a - b x^{2})^{2}} = \frac{3(a + b x^{2})}{(a - b x^{2})^{2}}
$$

### Final Answer
$$
f'(x) = \frac{3(a + b x^{2})}{(a - b x^{2})^{2}}
$$

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**Answer:** After simplifying, the derivative is 3(a + b x²) divided by (a – b x²) squared. Thus,

f ′(x) = 3(a + bx²)/(a − bx²)²
The derivative of $$ \frac{3x}{a - b x^{2}} $$ with respect to $$ x $$ is $$ \frac{3(a + b x^{2})}{(a - b x^{2})^{2}} $$.

2. Differentiate each of the following, (d). \( \frac{3 x}{a-b x^{2}} \)

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