We are tasked with using the quotient rule to find the derivative of the given function:

$y = \frac{x^2 - 3x + 5}{x + 7}$

Quotient Rule Formula:

If $y = \frac{u(x)}{v(x)}$, then the derivative $y'$ is given by:

$y' = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}$

In this case:

• $u(x) = x^2 - 3x + 5$
• $v(x) = x + 7$

We now need to find the derivatives of $u(x)$ and $v(x)$.

1. $u'(x) = \frac{d}{dx}(x^2 - 3x + 5) = 2x - 3$
2. $v'(x) = \frac{d}{dx}(x + 7) = 1$

Now applying the quotient rule:

$y' = \frac{(x + 7) \cdot (2x - 3) - (x^2 - 3x + 5) \cdot 1}{(x + 7)^2}$

Step-by-Step Solution:

1. Multiply terms in the numerator:

$(x + 7)(2x - 3) = 2x^2 + 14x - 3x - 21 = 2x^2 + 11x - 21$

2. Subtract the second part of the numerator:

$(2x^2 + 11x - 21) - (x^2 - 3x + 5) = 2x^2 + 11x - 21 - x^2 + 3x - 5 = x^2 + 14x - 26$

3. Final expression for the derivative:

$y' = \frac{x^2 + 14x - 26}{(x + 7)^2}$

This is the derivative of the given function using the quotient rule.

Would you like further explanation on any part of this process?

Related Questions:

1. How is the quotient rule different from the product rule?
2. Can you explain how the chain rule works when combined with the quotient rule?
3. What are some common mistakes when applying the quotient rule?
4. How does the derivative behave as $x \to -7$?
5. How would you apply the quotient rule to a trigonometric function?

Tip:

When applying the quotient rule, always ensure you correctly identify $u(x)$ and $v(x)$, and carefully differentiate each term to avoid mistakes.