Question

The function "f" is differentiable and (4f(t)+3t)dt over interval [0,x] = sin(x). Determine f'(π/6).

Answer 1

using
\[\int_{0}^{x} \ g(t) \ dt = G(x) - G(0)\]
meaning
\[\frac{d}{dx} \int_{0}^{x} \ g(t) \ dt = \frac{d}{dx} [G(x) - G(0)] = g(x)\]

we have:
\[\int_{0}^{x} \ 4f(t) + 3t \ dt = sin(x) \]
meaning
\[\frac{d}{dx} \int_{0}^{x} \ 4f(t) + 3t \ dt = \frac{d}{dx} sin(x) \]
and
\[4f(x) + 3x = cos x\]

from that, find \( f'(x)\) and then \( f'(\pi/6)\)