\(\rm \frac{d}{dx}x^{2x}\) =

Question

Question

Answer 1

Concept:

\(\rm log\ x^n=nlog\ x\)

\(\rm \frac{d}{dx}[log \ x ]=\frac1x\)

Product rule:

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

Calculation:

Let, y = x^2x

Taking log on both the sides, we get

log y = log( x2x ) = 2x logx

(∵ \(\rm log\ x^n=nlog\ x\))

Now, taking derivatives,

\(\rm \frac{d}{dx}[log \ y]\) = 2{ \(\rm \frac{d}{dx}[x] \ log \ x + \frac{d}{dx}[log \ x]\ x\) }

\(\rm \frac1y\frac{dy}{dx}\) = 2[log x + x. \(\rm \frac1x\)]

(∵ \(\rm \frac{d}{dx}[log \ x ]=\frac1x\))

⇒ \(\rm \frac{dy}{dx}\) = 2y [1 + log x]

⇒ \(\rm \frac{dy}{dx}\) = 2x2x [1 + log x]

(∵ y = x2x )

Hence, option (4) is correct.