https://file.aiquickdraw.com/mathbot/file/172686307606494thusbt.jpg

The extracted text from the image is slightly distorted, but based on the content, the function appears to be:

\[
f(x) = x\sqrt{5} + (\sqrt{5})^x + x^{\sqrt{5}}
\]

We need to find the derivative of this function and evaluate it at \(x = 1\).

Let's break this down and compute the derivative:

1. \( \frac{d}{dx} \left( x\sqrt{5} \right) = \sqrt{5} \)
2. \( \frac{d}{dx} \left( (\sqrt{5})^x \right) \) will use the exponential derivative formula, where \(a^x\) has the derivative \(a^x \ln(a)\), so:
   \[
   \frac{d}{dx} \left( (\sqrt{5})^x \right) = (\sqrt{5})^x \ln(\sqrt{5})
   \]
3. \( \frac{d}{dx} \left( x^{\sqrt{5}} \right) \) will use the power rule for fractional exponents, which gives:
   \[
   \frac{d}{dx} \left( x^{\sqrt{5}} \right) = \sqrt{5} x^{\sqrt{5} - 1}
   \]

Now, let's sum these up and evaluate at \(x = 1\).The derivative of the function \( f(x) = x\sqrt{5} + (\sqrt{5})^x + x^{\sqrt{5}} \) at \( x = 1 \) is approximately \( 6.27 \).

Would you like further details or explanations on this solution?

Here are five related questions to expand your understanding:
1. How would the derivative change for a different base in the exponential term?
2. Can you generalize the derivative for \( x^a \) where \( a \) is any real number?
3. How would the chain rule apply if the powers or bases were functions of \( x \)?
4. What are the common rules for differentiating functions involving constants like \( \sqrt{5} \)?
5. How would this derivative differ if we evaluated it at \( x = 2 \)?

**Tip**: When differentiating powers and exponentials, keep in mind the difference between the power rule \( \frac{d}{dx}(x^a) \) and the derivative of exponential functions like \( a^x \).

Find the derivative of f(x) = x√5 + (√5)^x + x^√5 when x = 1.

Learn how to find the derivative of the function f(x) = x√5 + (√5)^x + x^√5 at x = 1. This calculus problem covers key differentiation concepts, including the power rule, exponential derivative formula, and the chain rule. Suitable for college-level calculus students.

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