Question

find the derivative function of y=x^✓x​

Answer 1

Answer:

If y = x x and x > 0 then ln y = ln (x x)

Use properties of logarithmic functions to expand the right side of the above equation as follows.

ln y = x ln x

We now differentiate both sides with respect to x, using chain rule on the left side and the product rule on the right.

y '(1 / y) = ln x + x(1 / x) = ln x + 1 , where y ' = dy/dx

Multiply both sides by y

y ' = (ln x + 1)y

Substitute y by x x to obtain

y ' = (ln x + 1)x x

Step-by-step explanation:

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Answer 2

Answer :-

➩ $\sf y = {x}^{ \sqrt{x} }$

➩ $\sf y = x^{x^\frac{1}{2}}$

Rule used :-

Power rule - $\boxed{\sf\frac{d}{dx} x^n = nx^{n-1}}$

Solution :-

Differentiating with respect to x by using the above rule :-

➩ $\displaystyle\sf \dfrac{dy}{dx} = \dfrac{d(x^{x^\frac{1}{2}})}{dx}$

➩ $\displaystyle\sf \dfrac{dy}{dx} = x^\frac{1}{2} \times {(x)}^{1 -x^\frac{1}{2} }$

➩ $\displaystyle\boxed{\sf \dfrac{dy}{dx} = \sqrt{x} \times {(x)}^{ 1 - \sqrt{x}}}$