y = (x^0.5 − 6x − 7)(x^−1 + x^−2) solve dy/dx

We are tasked with finding the derivative \( \frac{dy}{dx} \) of the function:

\[
y = (x^{0.5} - 6x - 7)(x^{-1} + x^{-2})
\]

We will use the **product rule** and the **power rule** to differentiate this function. The product rule states that if \( y = f(x)g(x) \), then:

\[
\frac{dy}{dx} = f'(x)g(x) + f(x)g'(x)
\]

### Step 1: Define \( f(x) \) and \( g(x) \)
Let:
- \( f(x) = x^{0.5} - 6x - 7 \)
- \( g(x) = x^{-1} + x^{-2} \)

### Step 2: Differentiate \( f(x) \)
The derivative of \( f(x) = x^{0.5} - 6x - 7 \) is:

\[
f'(x) = \frac{1}{2}x^{-0.5} - 6
\]

### Step 3: Differentiate \( g(x) \)
The derivative of \( g(x) = x^{-1} + x^{-2} \) is:

\[
g'(x) = -x^{-2} - 2x^{-3}
\]

### Step 4: Apply the product rule
Now, applying the product rule:

\[
\frac{dy}{dx} = f'(x)g(x) + f(x)g'(x)
\]

Substitute \( f'(x) \), \( f(x) \), \( g'(x) \), and \( g(x) \):

\[
\frac{dy}{dx} = \left(\frac{1}{2}x^{-0.5} - 6\right)(x^{-1} + x^{-2}) + (x^{0.5} - 6x - 7)(-x^{-2} - 2x^{-3})
\]

### Step 5: Simplify
Now we expand both terms:

1. For \( \left(\frac{1}{2}x^{-0.5} - 6\right)(x^{-1} + x^{-2}) \):

\[
= \frac{1}{2}x^{-0.5}x^{-1} + \frac{1}{2}x^{-0.5}x^{-2} - 6x^{-1} - 6x^{-2}
\]
\[
= \frac{1}{2}x^{-1.5} + \frac{1}{2}x^{-2.5} - 6x^{-1} - 6x^{-2}
\]

2. For \( (x^{0.5} - 6x - 7)(-x^{-2} - 2x^{-3}) \):

\[
= x^{0.5}(-x^{-2}) + x^{0.5}(-2x^{-3}) - 6x(-x^{-2}) - 6x(-2x^{-3}) - 7(-x^{-2}) - 7(-2x^{-3})
\]
\[
= -x^{-1.5} - 2x^{-2.5} + 6x^{-1} + 12x^{-2} + 7x^{-2} + 14x^{-3}
\]

### Step 6: Combine all terms
Now, combine all the terms from the two expansions:

\[
\frac{dy}{dx} = \frac{1}{2}x^{-1.5} + \frac{1}{2}x^{-2.5} - 6x^{-1} - 6x^{-2} - x^{-1.5} - 2x^{-2.5} + 6x^{-1} + 12x^{-2} + 7x^{-2} + 14x^{-3}
\]

### Step 7: Simplify further
Group the like terms:

\[
\frac{dy}{dx} = \left(\frac{1}{2}x^{-1.5} - x^{-1.5}\right) + \left(\frac{1}{2}x^{-2.5} - 2x^{-2.5}\right) + (-6x^{-1} + 6x^{-1}) + (-6x^{-2} + 12x^{-2} + 7x^{-2}) + 14x^{-3}
\]

\[
= -\frac{1}{2}x^{-1.5} - \frac{3}{2}x^{-2.5} + 13x^{-2} + 14x^{-3}
\]

### Final Answer:

\[
\frac{dy}{dx} = -\frac{1}{2}x^{-1.5} - \frac{3}{2}x^{-2.5} + 13x^{-2} + 14x^{-3}
\]

Would you like to explore any part of this solution in more detail or have any questions?

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Here are five related questions for further exploration:

1. What is the interpretation of \( \frac{dy}{dx} \) in this context?
2. How would the solution change if there were additional terms in the original expression?
3. Can you factor the result of \( \frac{dy}{dx} \)?
4. How would you integrate the derivative \( \frac{dy}{dx} \)?
5. How does the product rule apply in cases with more than two terms?

**Tip:** Always break down complex expressions by applying fundamental rules like the product rule and chain rule step by step.

Learn how to differentiate the function y = (x^0.5 − 6x − 7)(x^−1 + x^−2) using the product rule and power rule. This solution provides a detailed step-by-step breakdown of the process, including the key calculus concepts and formulas involved. Suitable for students in Grades 11-12.

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