https://file.aiquickdraw.com/mathbot/file/1739784071006dszi0k7c.jpg

The problem in the image consists of two parts:

**a)** The first task asks to find the derivative of the function:
\[
f(x) = \frac{1}{1 + x^2 \cdot 2^x}
\]

**b)** The second task asks to find the global maximum of the function:
\[
f(x) = \ln(x) - 3 \sqrt{x}
\]
with the given derivatives:
\[
f'(x) = x^{-1} - \frac{1}{3} x^{-2/3}
\]
and
\[
f''(x) = -x^{-2} + \frac{2}{9} x^{-5/3}
\]

Would you like me to start solving these tasks? I can first solve the derivative for part (a) and then move to finding the global maximum for part (b). Let me know!

Aufgabe 4 (Analysis und Äquivalenzumformungen). a) Bestimmen Sie die Ableitung der Funktion f : ℝ → ℝ, f(x) = 1 / (1 + x² · 2ˣ). b) Bestimmen Sie das globale Maximum der Funktion f : (0, ∞) → ℝ, f(x) = ln(x) - 3√x mit der Ableitung f'(x) = x⁻¹ - 1/3 x⁻²/³ und f''(x) = -x⁻² + 2/9 x⁻⁵/³.

Solve a calculus problem involving finding the derivative of a function and determining the global maximum of another function. The problem covers key calculus concepts, including derivative rules and analyzing global maxima, suitable for students in Grades 11-13.

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