If
\[
f(x)=\frac1{x^2}\int_{3}^{x}\left(2t-3f'(t)\right)\,dt,
\]
then
\[
f'(3)=
\]
Show Hint
When an integral equation contains
\[
\int_a^x F(t)\,dt,
\]
first multiply away any outside factor and then differentiate using the Fundamental Theorem of Calculus:
\[
\frac{d}{dx}\int_a^x F(t)\,dt = F(x).
\]
Concept: Use the Fundamental Theorem of Calculus: differentiate after multiplying by \(x^2\) to remove the integral sign, then use the original equation to find \(f(3)\). Step 1: \(f(x) = \frac{1}{x^2}\int_3^x (2t-3f'(t))dt \Rightarrow x^2 f(x) = \int_3^x (2t-3f'(t))dt\). Step 2: Differentiate: \(2x f(x) + x^2 f'(x) = 2x - 3f'(x)\). Step 3: From original, \(f(3) = \frac{1}{9}\int_3^3 ... dt = 0\). Step 4: Put \(x=3\) into differentiated equation: \(2(3)(0) + 9f'(3) = 6 - 3f'(3) \Rightarrow 12f'(3)=6 \Rightarrow f'(3)=1/2\). Step 5: Write the final answer. \(\boxed{\frac12}\)