Question

Производная y=e^x * ln x

Answer 1

$y=e^x\cdot\ln x\\\\y'=(e^x)\cdot\ln x+e^x\cdot(\ln x)'=e^x\ln x+\frac{e^x}x$

Answer 2

To find the derivative of the function $y = e^x \ln x$, we will use the product rule of differentiation. The product rule states that if we have a function of the form $y = u(x) \cdot v(x)$, then its derivative is given by $y' = u'(x) \cdot v(x) + u(x) \cdot v'(x)$.

Let's proceed with finding the derivative:

Step 1: Identify $u(x)$ and $v(x)$: In our case, $u(x) = e^x$ and $v(x) = \ln x$.

Step 2: Find $u'(x)$ and $v'(x)$: The derivative of $e^x$ with respect to $x$ is simply $e^x$, and the derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$.

Step 3: Apply the product rule: $y' = u'(x) \cdot v(x) + u(x) \cdot v'(x) = e^x \cdot \ln x + e^x \cdot \frac{1}{x}$.

So, the derivative of $y = e^x \ln x$ with respect to $x$ is $y' = e^x \ln x + \frac{e^x}{x}$.