To differentiate the function 

\[
f(x) = \frac{5x^2 - 7x}{x^2 + 2},
\]

we will use the Quotient Rule, which states that if you have a function of the form:

\[
\frac{u(x)}{v(x)},
\]

then its derivative is given by:

\[
f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2},
\]

where \( u(x) = 5x^2 - 7x \) and \( v(x) = x^2 + 2 \).

Let's find \( u'(x) \) and \( v'(x) \):

1. \( u(x) = 5x^2 - 7x \)
   - \( u'(x) = 10x - 7 \)

2. \( v(x) = x^2 + 2 \)
   - \( v'(x) = 2x \)

Now, substitute into the Quotient Rule formula:

\[
f'(x) = \frac{(10x - 7)(x^2 + 2) - (5x^2 - 7x)(2x)}{(x^2 + 2)^2}.
\]

Let's simplify this step-by-step.The derivative of the function 

\[
f(x) = \frac{5x^2 - 7x}{x^2 + 2}
\]

is:

\[
f'(x) = \frac{7x^2 + 20x - 14}{x^4 + 4x^2 + 4}.
\]

This expression is fully simplified.

### Do you have any questions, or would you like a detailed breakdown of the steps?

### Related Questions:
1. How do you apply the Quotient Rule to other functions?
2. How does the simplification of derivatives work in more complex fractions?
3. What are the common mistakes when differentiating using the Quotient Rule?
4. Can this derivative be further simplified by factoring?
5. How would the process change if we used the Chain Rule instead?

### Tip:
Always simplify the derivative expression after applying the Quotient Rule to make it easier to interpret and use in further calculations.

Learn how to differentiate the function (5x^2 - 7x) / (x^2 + 2) using the Quotient Rule. This step-by-step solution covers key calculus concepts, including finding the derivative of rational functions. Suitable for Grades 11-12 students studying calculus.

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