f(x) = (3x - 6√x) / (5x^2 - 2) find the derivative

Steps to Solve

1. Identify the functions g(x) and h(x)

For our function $f(x) = \frac{3x - 6\sqrt{x}}{5x^2 - 2}$, we define:

• $g(x) = 3x - 6\sqrt{x}$ (the numerator)
• $h(x) = 5x^2 - 2$ (the denominator)

2. Calculate the derivatives g'(x) and h'(x)

We need to find the derivatives of both functions:

For $g(x)$:

• The derivative of $3x$ is $3$.
• The derivative of $-6\sqrt{x}$ can be calculated as $-6 \cdot \frac{1}{2}x^{-1/2} = -\frac{3}{\sqrt{x}}$.

Thus, $$ g'(x) = 3 - \frac{3}{\sqrt{x}} $$

For $h(x)$:

• The derivative of $5x^2$ is $10x$.
• The derivative of $-2$ is $0$.

So, $$ h'(x) = 10x $$

3. Apply the Quotient Rule

Now we apply the quotient rule for derivatives:

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

Plug in our values for $g(x)$, $g'(x)$, $h(x)$, and $h'(x)$:

$$ f'(x) = \frac{(3 - \frac{3}{\sqrt{x}})(5x^2 - 2) - (3x - 6\sqrt{x})(10x)}{(5x^2 - 2)^2} $$

4. Simplify the expression

Expand and simplify the numerator:

1. Multiply out $(3 - \frac{3}{\sqrt{x}})(5x^2 - 2)$:
   • $3 \cdot (5x^2 - 2) = 15x^2 - 6$
   • $-\frac{3}{\sqrt{x}} \cdot (5x^2 - 2) = -\frac{15x^{3/2}}{\sqrt{x}} + \frac{6}{\sqrt{x}}$

So the full numerator becomes: $$ 15x^2 - 6 - \left( 30x^2 - 60\sqrt{x} \right) $$

Combine like terms to simplify.

5. Final result

This gives us the final expression for $f'(x)$ which can be simplified further to express it clearly.