Question

find the derivative of y with respect to x if y = ln x^1/2
the derivative of y with respect to x of y = ln x^1/2 is

Explanation:

Step1: Recall derivative formula

The derivative of $\ln(u)$ with respect to $x$ is $\frac{u'}{u}$ by the chain - rule. Here $u = x^{\frac{1}{2}}$.

Step2: Find the derivative of $u = x^{\frac{1}{2}}$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $n=\frac{1}{2}$, we have $u'=\frac{1}{2}x^{-\frac{1}{2}}$.

Step3: Calculate the derivative of $\ln(x^{\frac{1}{2}})$

Substitute $u = x^{\frac{1}{2}}$ and $u'=\frac{1}{2}x^{-\frac{1}{2}}$ into $\frac{u'}{u}$. So $\frac{d}{dx}\ln(x^{\frac{1}{2}})=\frac{\frac{1}{2}x^{-\frac{1}{2}}}{x^{\frac{1}{2}}}$.

Step4: Simplify the expression

$\frac{\frac{1}{2}x^{-\frac{1}{2}}}{x^{\frac{1}{2}}}=\frac{1}{2}x^{-\frac{1}{2}-\frac{1}{2}}=\frac{1}{2}x^{-1}=\frac{1}{2x}$.

Answer:

$\frac{1}{2x}$