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 \( \left(6, \frac{1}{6}\right) \), we first need to determine the derivative of \( y \). The derivative \( y' = -\frac{1}{x^2} \) gives us the slope of the tangent line. At \( x = 6 \), the slope is \( y' = -\frac{1}{36} \). Now, using the point-slope form of a line, the equation becomes: \[ y - \frac{1}{6} = -\frac{1}{36}(x - 6) \] Simplifying this, we find: \[ y = -\frac{1}{36}x + \frac{1}{6} + \frac{1}{6} = -\frac{1}{36}x + \frac{2}{6} = -\frac{1}{36}x + \frac{1}{3} \] So, the equation of the tangent line is: \[ y = -\frac{1}{36}x + \frac{1}{3} \] It’s definitely a fun exercise to visualize curves and tangents! Did you know the concept of tangents has been around since ancient Greeks, with scholars like Euclid and Archimedes exploring shapes and lines? Fast forward to today, and we rely heavily on derivatives in calculus for everything from physics to economics, allowing us to analyze change and motion, making it a crucial skill!