1.

Find the first order derivative of  x log x1. 1 + log x2. log x + x3. x log x4. None of these

Answer» Correct Answer - Option 1 : 1 + log x

Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

​Product Rule: 

\(\rm \dfrac {d}{dx} [f(x)g(x)] = f(x) \dfrac {d}{dx} g(x) + g(x) \dfrac {d}{dx} f(x)\)

\(\rm \dfrac {d}{dx} (\log x) = \dfrac {1}{x}\)

 

Calculation:

Consider, y = x log x

Taking derivative w. r. to x on both side, we get

⇒ \(\rm \dfrac {dy}{dx} = \dfrac {d}{dx} (x \log x)\)

⇒\(\rm \dfrac {dy}{dx} = x \dfrac {d}{dx} (\log x) + (\log x) \dfrac {d}{dx} (x)\)

⇒\(\rm \dfrac {dy}{dx} = x \;(\dfrac {1}{x} )+ \log x \;(1)\)

⇒\(\rm \dfrac {dy}{dx} = 1+ \log x \)

 Hence,  the first order derivative of  x log x is \(\rm 1+ \log x \)


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