Differentiate

ln
⁡
(
7
𝑥
+
5
)

Question

Differentiate

ln
⁡
(
7
𝑥
+
5
)

BlackTom AI · Accepted Answer

To differentiate the natural logarithm of a function, use the chain rule: if y = ln(u(x)), then dy/dx = u'(x)/u(x).

Here, u(x) = 7x + 5. Its derivative is u'(x) = 7.

Applying the rule, the derivative is dy/dx = 7 / (7x + 5).

In a typical multiple-choice setup, you would look for the option that matches 7/(7x+5). The provided answer field in this item shows '7(7x+5)' which is incorrect dimensionally (it would imply multiplying by the denominator rather than dividing by it). Moreover, the answer options array is empty, so there is no alternative choice to evaluate.

Key points to keep in mind:
- The derivative of ln(v) is v'/v, not v'.
- The chain rule introduces the inner derivative (the 7 from 7x) in the numerator, resulting in 7/(7x+5).
- If you were to simplify or manipulate, remember that 7/(7x+5) cannot be reduced further in terms of elementary algebraic simplification.

If an option list were present, you would compare each candidate against 7/(7x+5) and discard any that misapply the rule (e.g., ln(7x+5) becomes (7)/(7x+5) for the derivative, whereas ln(x) would yield 1/x, and constants outside the logarithm would not disappear from the numerator).

Differentiate ln ⁡ ( 7 𝑥 + 5 )

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