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

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

What is the Derivative of x^lnx?

Answer:

Explanation:

To find the derivative of x^lnx, we will use the logarithmic differentiation. Let us put

y = x^lnx.

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

ln y = ln x^lnx

⇒ 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

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

So the derivative of x^lnx is equal to 2x^{lnx -1} lnx.

More Derivatives: Derivative of 1/lnx

Derivative of ln(lnx)

Derivative of sinx/x

Derivative of 1/sin x

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