Tangent lines

Problem 2.912 · medium

1. The tangent line passes through the point (a, f(a)) and has slope f'(a).
   Reviewed
2. \[ \left. - x^{2} + 4 x + 3 + \frac{3}{x} \right|_{\substack{ x=1 }} = 9 \]
   The point of tangency.✓ Proved
3. \[ \frac{d}{d x} \left(- x^{2} + 4 x + 3 + \frac{3}{x}\right) = - 2 x + 4 - \frac{3}{x^{2}} \]
   Differentiate.✓ Proved
4. \[ \left. - 2 x + 4 - \frac{3}{x^{2}} \right|_{\substack{ x=1 }} = -1 \]
   The slope at the point.✓ Proved
5. \[ 10 - x \]
   Point-slope form, then simplify.✓ Proved