Derivative of x^lnx [x to the power lnx]

The derivative of x^lnx (x to the power lnx) is equal to 2xlnx -1 lnx. Here, ln denotes the natural logarithm, that is, lnx =loge x. In this post, we will learn how to differentiate xlnx.

What is the Derivative of xlnx?



To find the derivative of xlnx, we will use the logarithmic differentiation. Let us put

y = xlnx.

We need to find dy/dx. Taking logarithms both sides, we get that

ln y = ln xlnx

⇒ ln y = lnx lnx

⇒ ln y = (lnx)2

Differentiating with respect to x, we obtain that

$\dfrac{1}{y} \dfrac{dy}{dx}= 2 \ln x \dfrac{d}{dx}(\ln x)$, by the chain rule of derivatives.

⇒ $\dfrac{1}{y} \dfrac{dy}{dx}= 2 (\ln x) \dfrac{1}{x}$

⇒ $\dfrac{dy}{dx}= 2 y (\ln x)$ x-1

⇒ $\dfrac{dy}{dx}= 2\ln x$xlnx-1 as y=xlnx.

So the derivative of xlnx is equal to 2xlnx -1 lnx.

More Derivatives: Derivative of 1/lnx

Derivative of ln(lnx)

Derivative of sinx/x

Derivative of 1/sin x


Q1: What is the differentiation of xlnx?

Answer: The differentiation of xlnx is equal to 2xlnx -1 lnx.

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