find the second derivative of f (x)= 5x-7/x²-8

To find the second derivative of the function \( f(x) = \frac{5x - 7}{x^2 - 8} \), we'll follow these steps:

### Step 1: Find the First Derivative

First, apply the quotient rule. The quotient rule states that if \( f(x) = \frac{g(x)}{h(x)} \), then:

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

Here, \( g(x) = 5x - 7 \) and \( h(x) = x^2 - 8 \).

1. Differentiate \( g(x) \) with respect to \( x \):

\[
g'(x) = 5
\]

2. Differentiate \( h(x) \) with respect to \( x \):

\[
h'(x) = 2x
\]

3. Apply the quotient rule:

\[
f'(x) = \frac{(5)(x^2 - 8) - (5x - 7)(2x)}{(x^2 - 8)^2}
\]

Simplify the numerator:

\[
f'(x) = \frac{5x^2 - 40 - (10x^2 - 14x)}{(x^2 - 8)^2}
\]

Combine like terms:

\[
f'(x) = \frac{-5x^2 + 14x - 40}{(x^2 - 8)^2}
\]

### Step 2: Find the Second Derivative

Now, differentiate \( f'(x) \) to find \( f''(x) \).

This requires applying the quotient rule again. Let:

\[
u(x) = -5x^2 + 14x - 40, \quad v(x) = (x^2 - 8)^2
\]

1. Differentiate \( u(x) \):

\[
u'(x) = -10x + 14
\]

2. Differentiate \( v(x) \) using the chain rule:

\[
v'(x) = 2(x^2 - 8) \cdot 2x = 4x(x^2 - 8)
\]

3. Apply the quotient rule to find \( f''(x) \):

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

Substituting \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the formula:

\[
f''(x) = \frac{(-10x + 14)(x^2 - 8)^2 - (-5x^2 + 14x - 40) \cdot 4x(x^2 - 8)}{(x^2 - 8)^4}
\]

Simplifying this expression involves expanding the terms, combining like terms, and reducing the fraction.

Would you like the detailed expansion and final simplified expression for the second derivative? 

### 5 Related Questions
1. How do you use the product rule in calculus?
2. What are the applications of the second derivative in real-life problems?
3. Can you explain the chain rule for differentiation with an example?
4. How does implicit differentiation work for functions defined implicitly?
5. What are the conditions for a function to be concave or convex?

**Tip:** The second derivative helps determine the concavity of a function, which can indicate the presence of local maxima or minima.

Learn how to find the second derivative of the function f(x) = (5x - 7)/(x^2 - 8) step-by-step. This problem involves differentiation using the quotient rule and finding higher-order derivatives. Suitable for advanced calculus students.

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