Differentiate

ln
⁡
(
𝑥
2
−
5
2
𝑥
+
1
)

Question

Differentiate

ln
⁡
(
𝑥
2
−
5
2
𝑥
+
1
)

BlackTom AI · Accepted Answer

The question asks to differentiate the natural logarithm of a quadratic expression. First, identify the inner function: f(x) = x^2 − (5/2)x + 1. The derivative of ln(f(x)) with respect to x is f'(x) / f(x) by the chain rule.

Compute f'(x): the derivative of x^2 is 2x, and the derivative of −(5/2)x is −5/2, while the constant 1 vanishes. So f'(x) = 2x − 5/2.

Form the fraction: d/dx [ln(f(x))] = (2x − 5/2) / (x^2 − (5/2)x + 1).

For a cleaner form, you can multiply numerator and denominator by 2 to clear the fractions: d/dx [ln(f(x))] = (4x − 5) / (2x^2 − 5x + 2).

At this point, you would typically compare with the provided answer choices to select the correct one. However, the given data contains no answer options to evaluate against. The derivative above represents the correct differentiated form of ln(x^2 − (5/2)x + 1).

If you intended a specific alternative form (for example integrating constant factors or factoring the denominator), you could re-express (4x − 5) / (2x^2 − 5x + 2) in equivalent fractional forms, but any equivalent expression must be algebraically identical to this result. Without the answer options, we cannot determine which, if any, matches a provided choice.

Differentiate ln ⁡ ( 𝑥 2 − 5 2 𝑥 + 1 )

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