How do you differentiate #e^(lnx) #?

Question

Emma Aldridge · Accepted Answer

Use #e^lnx = x#.

Cameron Alexander · Answer

undefined To differentiate $ e^{\ln(x)} $, you can use the chain rule. The derivative is:

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

Using the chain rule and the derivative of $\ln(x)$ with respect to $x$, which is $1/x$, the expression simplifies to:

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

Thus, the derivative of $e^{\ln(x)}$ is $ \frac{1}{x} \cdot e^{\ln(x)} $, or simply $ \frac{e^{\ln(x)}}{x} $.

Isabella Ackerman · Answer

undefined To differentiate $ e^{\ln(x)} $, we can use the chain rule. The chain rule states that if we have a function $ f(g(x)) $, then its derivative is given by $ f'(g(x)) \times g'(x) $.

In this case, $ f(g(x)) = e^{\ln(x)} $ and $ g(x) = \ln(x) $.

The derivative of $ e^{\ln(x)} $ can be found as follows:

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

Now, we need to find the derivative of $ \ln(x) $. The derivative of $ \ln(x) $ is $ \frac{1}{x} $.

Substituting this into our expression, we get:

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

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

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