# Find the slope at any value of x,if y=(x)^lnx?

May 27, 2018

The slope at any value of $x$ is given by ${x}^{\ln} x \left(\frac{2}{x} \ln x\right)$

#### Explanation:

This question is pretty much asking for the derivative of $y$. We will use logarithmic differentiation for this, aka take the logarithm of both sides of the function.

$\ln y = \ln \left({x}^{\ln} x\right)$

Now apply the logarithm property that states $\ln \left({a}^{n}\right) = n \ln a$.

$\ln y = \ln x \ln x$

$\frac{1}{y} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{1}{x} \ln x + \frac{1}{x} \ln x$

$\frac{1}{y} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{2}{x} \ln x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = y \left(\frac{2}{x} \ln x\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{\ln} x \left(\frac{2}{x} \ln x\right)$

Hopefully this helps!