Mary Karanja Find the equation of the tangent line to the curve \( y=\frac{1}{x} \) at \( \left(6, \frac{1}{6}\right) \) Equation

To find the equation of the tangent line to the curve \( y = \frac{1}{x} \) at the point \( \left(6, \frac{1}{6}\right) \), follow these steps: ### Step 1: Find the Derivative The derivative of a function gives the slope of the tangent line at any point on the curve. Given: \[ y = \frac{1}{x} \] Rewrite \( y \) as: \[ y = x^{-1} \] Differentiate with respect to \( x \): \[ \frac{dy}{dx} = -x^{-2} = -\frac{1}{x^2} \] ### Step 2: Determine the Slope at \( x = 6 \) Evaluate the derivative at \( x = 6 \) to find the slope (\( m \)) of the tangent line at that point. \[ m = \left. \frac{dy}{dx} \right|_{x=6} = -\frac{1}{6^2} = -\frac{1}{36} \] ### Step 3: Use the Point-Slope Formula The point-slope form of a line is: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point of tangency. Given the point \( \left(6, \frac{1}{6}\right) \) and \( m = -\frac{1}{36} \), plug these into the formula: \[ y - \frac{1}{6} = -\frac{1}{36}(x - 6) \] ### Step 4: Simplify to Slope-Intercept Form Expand and simplify the equation to get it into the slope-intercept form \( y = mx + b \): \[ \begin{align*} y - \frac{1}{6} &= -\frac{1}{36}x + \frac{6}{36} \\ y - \frac{1}{6} &= -\frac{1}{36}x + \frac{1}{6} \\ y &= -\frac{1}{36}x + \frac{1}{6} + \frac{1}{6} \\ y &= -\frac{1}{36}x + \frac{2}{6} \\ y &= -\frac{1}{36}x + \frac{1}{3} \end{align*} \] ### Final Equation The equation of the tangent line is: \[ y = -\frac{1}{36}x + \frac{1}{3} \] --- **Final Answer:** After simplifying, an equation for the tangent line is \[ y = -\frac{1}{36}\,x + \frac{1}{3} \]