Question

Найти производную f(x)=e^lgx

Answer 1

$y=e^{\lg x}\\y'=e^{\lg x}\cdot(\lg x)'=e^{\lg x}\cdot\dfrac{1}{x\ln10} =\dfrac{e^{\lg x}}{x\ln10}$

Answer 2

To find the derivative of the function $f(x) = e^{\log{x}}$, we will use the chain rule. First, let's express $f(x)$ in a simpler form by noting that $\log{x} = \ln{x}$ (natural logarithm). So, we have:

$f(x) = e^{\ln{x}}$

Now, using the chain rule, the derivative of $f(x)$ with respect to $x$ is given by:

$\frac{d}{dx} \left(e^{\ln{x}}\right) = e^{\ln{x}} \cdot \frac{d}{dx} (\ln{x})$

Now, let's calculate $\frac{d}{dx} (\ln{x})$. The derivative of the natural logarithm $\ln{x}$ with respect to $x$ is simply $1/x$:

$\frac{d}{dx} (\ln{x}) = \frac{1}{x}$

Substituting this back into our previous equation:

$\frac{d}{dx} \left(e^{\ln{x}}\right) = e^{\ln{x}} \cdot \frac{1}{x}$

Since $e^{\ln{x}} = x$, the derivative becomes:

$\frac{d}{dx} \left(e^{\ln{x}}\right) = \frac{x}{x} = 1$

So, the derivative of the function $f(x) = e^{\log{x}}$ with respect to $x$ is simply $1$.