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

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

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Answer 1 · 2018-05-27T03:48:13

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

Explanation:

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

$ln y = ln (x^{ln x})$ $lny = ln(x^lnx)$

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

$ln y = ln x ln x$ $lny = lnxlnx$

$\frac{1}{y} (\frac{d y}{d x}) = \frac{1}{x} ln x + \frac{1}{x} ln x$ $1/y(dy/dx) = 1/xlnx + 1/xlnx$

$\frac{1}{y} (\frac{d y}{d x}) = \frac{2}{x} ln x$ $1/y(dy/dx) = 2/xlnx$

$\frac{d y}{d x} = y (\frac{2}{x} ln x)$ $dy/dx = y(2/xlnx)$

$\frac{d y}{d x} = x^{ln x} (\frac{2}{x} ln x)$ $dy/dx= x^lnx(2/xlnx)$

Hopefully this helps!

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

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