Question

If f(x) = x (ln x), find f'(x).

A. e^(x + 1)
B. 1 + ln x
C. 1 + e^x
D. (ln x)^2 + x

Answer 1

`f'(x)=1+\ln x`

Explanation:

`f'(x)` is notation for the first derivative of
`f(x)`.

Recall the product rule:

`(f\cdot g)'=f'\cdot g+g'\cdot f`

Therefore, we have:

`\displaystyle (d)/(dx)(x\ln x)=(d)/(dx)(x)\cdot \ln(x)+(d)/(dx)(\ln x)\cdot x`

Note that:

`\displaystyle (d)/(dx)(x)=1,\\\\(d)/(dx)(\ln x)=(1)/(x)`

Simplify:

`\displaystyle (d)/(dx)(x\ln x)=1\cdot \ln x+(1)/(x)\cdot x, \\\\(d)/(dx)(x\ln x)= \ln x+1=\boxed{1+\ln x}`

Answer 2

The derivative is:

`\bold{f(x) \: = \: x \: * \: (In(x))}`

`\bold{f'(x) \: = \: (d)/(dx) (x \: * \: In(x))}`

`\bold{f'(x) \: = \: (d)/(dx) (x) \: * \: In(x) \: + \: x \: * \: (d)/(dx) (In(x))}`

`\bold{f'(x) \: = \: 1In(x) \: + \: x \: * (1)/(x) }`

`\bold{f'(x) \: = \: In(x) \: + \: x \: * \: (1)/(x) }`

`\boxed{ \bold{f'(x) \: = \: In(x) \: + \: 1}}`

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