If F(x) = 13x/(4 + x2), find F '(3) and use it to find an equation of the tangent line to the curve y = 13x/(4 + x2) at the point (3, 3).

Question

If F(x) = 13x/(4 + x2), find F '(3) and use it to find an equation of the tangent line to the curve y = 13x/(4 + x2) at the point (3, 3).

F'(3)=

y(x)=

Answer 1

The equation of the tangent line to F at the point (3,3) is:

y = F'(3)(x - 3) + 3

So, we need to find F'(3). Using the quotient rule, we have:

F'(x) = [13(4 + x²) - 13x(2x)]/(4 + x²)² = (-13x² + 52)/(x² + 4)²

Plugging in x = 3, we have:

F'(3) = (-13(9) + 52)/13² = (-117 + 52)/169 = -65/169

Therefore, the equation of the tangent line is:

y = (-65/169)(x - 3) + 3