Question

If f(x) = x² - 2x, then d/dx(f(lnx)) is:

a) (2/x) - 2
b) (2/x) + 2
c) (2/x)
d) (2/x) - 1

Answer 1

If f(x) = x² - 2x, then d/dx(f(lnx)) is: 2/x. Thus, the correct answer is option c) (2/x).

Step-by-step explanation:

The derivative
`\((d/dx)\) of \(f(\ln(x))\)` can be found by applying the chain rule. Let
`\(u = \ln(x)\)`, then
`\(f(\ln(x))\)` becomes
`\(f(u) = u^2 - 2u\)`.

Now, applying the chain rule, the derivative is given by
`\((d/du)(u^2 - 2u) \cdot (d/dx)(\ln(x))\)`.

The derivative of
`\(u^2 - 2u\)` with respect to u is 2u - 2.

The derivative of
`\(\ln(x)\)` with respect to x is
`\(1/x\).` Multiplying these, we get
`\((2u - 2) \cdot (1/x)\)`.

Substitute back
`\(u = \ln(x)\)`, and we have
`\((2\ln(x) - 2) / x\)`.

To simplify further, factor out a 2 from the numerator to get

`\(2(\ln(x) - 1) / x\)`.

Now, distribute the 2 in the numerator to get
`\((2\ln(x) - 2) / x\)`.

Finally, simplify by canceling the common factor of 2, resulting in the final answer
`\((2\ln(x) - 2) / x\)`, which can be expressed as
`\(2/x - 2/x\)`. Therefore, the correct answer is option c) (2/x)