Compute the first derivative of
𝑓
(
𝑥
)
=
64
+
30
𝑥
0.5
−
𝑥
−
2

Question

Compute the first derivative of 
𝑓
(
𝑥
)
=
64
+
30
𝑥
0.5
−
𝑥
−
2

BlackTom AI · Accepted Answer

Using the power rule d/dx x^n = n x^{n-1} immediately gives each term's derivative.

1) Relevant knowledge points and concepts
- Constant rule: derivative of a constant is 0.
- Constant multiple rule: derivative of c·g(x) is c·g'(x).
- Power rule: for any real n, d/dx x^n = n x^{n-1}.
- Domain note: x^{0.5} requires x ≥ 0; x^{-2} requires x ≠ 0, so for real-valued function and derivative consider x > 0.

2) Step-by-step reasoning and calculations
Start with f(x) = 64 + 30 x^{0.5} − x^{−2}.

- Derivative of 64: d/dx(64) = 0.

- Derivative of 30 x^{0.5}:
  write exponent as 1/2. Apply power rule with constant multiple:
  d/dx(30 x^{1/2}) = 30 · (1/2) x^{1/2 − 1}
  = 30 · (1/2) x^{−1/2}
  = 15 x^{−1/2}.

- Derivative of − x^{−2}:
  d/dx(− x^{−2}) = − (−2) x^{−2 − 1}
  = 2 x^{−3}.

3) Combine the results
f′(x) = 0 + 15 x^{−1/2} + 2 x^{−3}
so f′(x) = 15 x^{−1/2} + 2 x^{−3}.

4) Why this is correct
Each term follows directly from the power rule and constant multiple rule; signs and exponents were handled by subtracting 1 from each exponent and multiplying by the original exponent. No additional rules (chain rule, product rule) are needed.

5) Why common alternatives are incorrect
- Forgetting to multiply by the original exponent (e.g., writing 30 x^{−1/2} instead of 15 x^{−1/2}) misses the 1/2 factor.
- Dropping the sign change on −x^{−2} (giving −2 x^{−3}) would be incorrect sign algebra.
- Treating x^{0.5} derivative as 0 would ignore the power rule.

6) Conclusion
Thus the derivative is f′(x) = 15 x^{−1/2} + 2 x^{−3}, which matches the provided correct answer.

Compute the first derivative of 𝑓 ( 𝑥 ) = 64 + 30 𝑥 0.5 − 𝑥 − 2Single choice

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