Equations of tangent lines by definition (2)
b. Determine an equation of the tangent line at P.
f(x) = √x+3; P (1,2)

Verified step by step guidance

1. Step 1: Understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y_1 = m(x - x_1), where m is the slope of the tangent line, and (x_1, y_1) is the point of tangency.
2. Step 2: Identify the function f(x) = \sqrt{x} + 3 and the point P(1, 2). Here, x_1 = 1 and y_1 = 2.
3. Step 3: To find the slope m of the tangent line, calculate the derivative of the function f(x). The derivative f'(x) represents the slope of the tangent line at any point x.
4. Step 4: Differentiate f(x) = \sqrt{x} + 3. The derivative f'(x) = \frac{1}{2\sqrt{x}}. This is because the derivative of \sqrt{x} is \frac{1}{2\sqrt{x}}, and the derivative of a constant is 0.
5. Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at P. Substitute x = 1 into f'(x) to get m = f'(1).