Question

Производная от ln^2 x

Answer 1

$(\ln^2x)'=2\ln x*\frac{1}{x}= \frac{2 \ln x}{x}$

Answer 2

To find the derivative of the function $f(x) = (\ln x)^2$, we will use the chain rule and the power rule of differentiation.

The chain rule states that if we have a function $g(u)$ where $u$ is a function of $x$, then the derivative of $g(u)$ with respect to $x$ is $g'(u) \cdot u'$.

The power rule states that if we have a function $h(x) = x^n$, then its derivative is $h'(x) = n \cdot x^{n-1}$.

Let's apply these rules to find the derivative of $f(x)$:

$f(x) = (\ln x)^2$

Using the chain rule, we let $u = \ln x$ and $g(u) = u^2$:

$g'(u) = 2u$ (derivative of $u^2$ with respect to $u$)

Now, we need to find $u'$, which is the derivative of $\ln x$:

$u = \ln x$

Using the power rule, $u' = \frac{1}{x}$ (derivative of $\ln x$ with respect to $x$)

Now, we can combine everything to find the derivative of $f(x)$:

$f'(x) = g'(u) \cdot u'$

$f'(x) = 2u \cdot \frac{1}{x}$

$f'(x) = \frac{2 \ln x}{x}$

So, the derivative of $(\ln x)^2$ with respect to $x$ is $\frac{2 \ln x}{x}$.