What is the Derivative of x^logx

The derivative of x^logx (x to the power logx) is equal to 2 logx xlogx-1. Here we find the differentiation of xlogx by using the product rule of derivatives together with logarithmic differentiation.

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The product rule of derivatives states the following:

$\dfrac{d}{dx}$(f(x)g(x)) = f(x) $\dfrac{d}{dx}$(g(x)) + g(x) $\dfrac{d}{dx}$(f(x)) …(I)

Derivative of x^logx

Find the Derivative of xlogx

Answer: The derivative of xlog x is 2xlogx-1 logx.

We find the derivative of xlogx as described below. Here we have considered natural logarithms, that is, logx = loge x.

Put y=xlogx …(II)

To find $\frac{dy}{dx}$, we use the method of logarithmic differentiation. Taking logarithms on both sides of (II) we get that

loge y = loge xlogx

⇒ log y = logx log x, here we have used the logarithm rule: log am = m log a.

Differentiating both sides we obtain that

$\dfrac{1}{y}  \dfrac{dy}{dx}=\dfrac{d}{dx}$(logx logx)

⇒ $\dfrac{1}{y}  \dfrac{dy}{dx}$ = logx $\dfrac{d}{dx}$(logx) + logx $\dfrac{d}{dx}$(logx) using the above product rule (I) of derivatives.

⇒ $\dfrac{1}{y}  \dfrac{dy}{dx}$ = logx ⋅ $\dfrac{1}{x}$ + logx ⋅ $\dfrac{1}{x}$

⇒ $\dfrac{dy}{dx}$ = y [$\dfrac{2 \log x}{x}$]

⇒ $\dfrac{dy}{dx}$ = xlogx [$\dfrac{2 \log x}{x}$] as y=xlogx.

⇒ $\dfrac{dy}{dx}$ = 2 logx xlogx-1

So the derivative of xlogx is equal to 2 logx xlogx-1.

Related Derivatives:

Derivative of (sinx)logxDerivative of xx
Derivative of sinx/xDerivative of xsinx


Q1: What is the derivative of xlogx?

Answer: The derivative of xlogx (x raised to the logx) is equal to 2xlogx-1 logx.

Q2: If y= xlogx , then find dy/dx.

Answer: If y=xlogx, then dy/dx= 2 logx xlogx-1.

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