f(x) = 8x4 - 3x-7 + x - 23    

Find f’(x)

Let's first rewrite the function \( f(x) \) in its simplified form:

\[
f(x) = 8x^4 - 3x^{-7} + x - 23
\]

Now, we will differentiate each term with respect to \( x \) using standard differentiation rules:

- The derivative of \( 8x^4 \) is \( 32x^3 \) (since the power rule states \( \frac{d}{dx}(x^n) = nx^{n-1} \)).
- The derivative of \( -3x^{-7} \) is \( 21x^{-8} \).
- The derivative of \( x \) is \( 1 \).
- The derivative of a constant \( -23 \) is \( 0 \).

Thus, the derivative \( f'(x) \) is:

\[
f'(x) = 32x^3 + 21x^{-8} + 1
\]

Let me know if you'd like further clarification or details.

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Here are 5 related questions to explore this topic further:
1. What are the rules for differentiating polynomial functions?
2. How does the power rule for differentiation work for negative exponents?
3. Can you apply the chain rule to a function like \( f(x) = (8x^4 - 3x^{-7})^2 \)?
4. What is the derivative of \( g(x) = 5x^{-3} + 4x^2 - 7x \)?
5. How would the second derivative \( f''(x) \) of \( f(x) = 8x^4 - 3x^{-7} + x - 23 \) be found?

**Tip:** When differentiating functions with negative exponents, always reduce the exponent by 1, just like with positive exponents, but watch for signs!

Find the derivative of f(x) = 8x^4 - 3x^{-7} + x - 23.

Learn how to differentiate the polynomial function f(x) = 8x^4 - 3x^{-7} + x - 23 using the power rule for differentiation. This problem involves higher-degree terms and negative exponents, providing a step-by-step solution suitable for Grades 11-12.

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