Questions: Find f'(x) and find the equation of the line tangent to the graph of f at x = 3, where f(x) = (4x - 3)^(1/2) f'(x) = The equation of the tangent line is y = A. The tangent line is horizontal at x = 3. B. The tangent line is never horizontal.

Solution Steps

To find the derivative of the function \( f(x) = (4x - 3)^{1/2} \), we apply the chain rule. The derivative is given by: \[ f'(x) = \frac{2}{\sqrt{4x - 3}} \]

Next, we evaluate the derivative at \( x = 3 \) to find the slope of the tangent line: \[ f'(3) = \frac{2}{\sqrt{4(3) - 3}} = \frac{2}{\sqrt{9}} = \frac{2}{3} \]

We also need to find the value of the function at \( x = 3 \): \[ f(3) = \sqrt{4(3) - 3} = \sqrt{9} = 3 \] Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m = \frac{2}{3} \) and \( (x_1, y_1) = (3, 3) \), we can write the equation of the tangent line: \[ y - 3 = \frac{2}{3}(x - 3) \] Simplifying this, we find: \[ y = \frac{2}{3}x + 1 \]

Final Answer