Differentiate

[
ln
⁡
(
3
𝑥
)
+
5
]
2

Question

Differentiate

[
ln
⁡
(
3
𝑥
)
+
5
]
2

BlackTom AI · Accepted Answer

Let's restate what we’re differentiating and what the given answer appears to be.
The problem asks for the derivative of the expression [ln(3x) + 5]^2 with respect to x. Here, the inner function is u(x) = ln(3x) + 5, and the outer function is v(u) = u^2.
Applying the chain rule: d/dx [u(x)^2] = 2 u(x) · u'(x).
Now compute u'(x): derivative of ln(3x) with respect to x is (1/(3x)) · 3 by the chain rule, which simplifies to 1/x. The derivative of +5 is 0. Therefore u'(x) = 1/x.
Putting it together: d/dx [ln(3x) + 5]^2 = 2 [ln(3x) + 5] · (1/x) = 2[ln(3x) + 5]/x.
The provided answer expression is 2 [ ln(3x) + 5 ] x. This would imply multiplying by x instead of dividing by x, which does not match the correct application of the chain rule and the derivative of ln(3x). The discrepancy arises from a misplacement of the x in the denominator; the correct form must divide by x, not multiply by x.
If there were additional answer options, we would compare each one: for any option that matches 2[ln(3x) + 5]/x, that would be a correct form; any option that incorrectly uses multiplication by x, or omits the 2, or omits the 1/x factor, would be incorrect. In this case, only the single provided option is present, and it does not align with the proper derivative, pointing to an incorrect result due to the missing division by x.

Differentiate [ ln ⁡ ( 3 𝑥 ) + 5 ] 2

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