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What is the derivative of (9x³)³ˣ?

a) 27x(9x³)³ˣ⁻¹
b) (9x³)³ˣ(3 ln(9x³)-1/3x
c) (9x³)³ˣ(3 ln(9x³) - 9x³ˣ⁻¹)
d) 3x(9x³)³ˣ⁻¹
e) 3 ln(9x³) - 9x³ˣ⁻¹)

1 Answer

7 votes

Final answer:

The derivative of (9x³)³ˣ requires using the chain rule and the exponential rule. The correct derivative is (9x³)^{3x} · (3 ln(9x³) + 81x²), but this does not match any of the provided options, suggesting a possible error in the question or answer choices.

Step-by-step explanation:

The student has asked about the derivative of the function (9x³)³ˣ. To find this, we will have to use the chain rule and the exponential rule for derivatives. Let's denote the function as f(x) = (g(x))^{h(x)} where g(x) = 9x³ and h(x) = 3x. The derivative of a function of this form is given by:

f'(x) = h(x) · g(x)^{h(x)-1} · g'(x) + g(x)^{h(x)} · ln(g(x)) · h'(x)

Applying this to our function gives:

f'(x) = 3x · (9x³)^{3x-1} · 27x² + (9x³)^{3x} · ln(9x³) · 3

After simplifying, we get the correct answer:

f'(x) = (9x³)^{3x} · (3 ln(9x³) + 81x²)

However, none of the provided options exactly matches this result. The student may need to recheck the problem or the given answer choices, as there might have been a typo or mistake in the transcription of the available options.

User Gboda
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