Find the slope of the tangent Line given point. b) \( g(x)=\frac{1}{x} ;(1, x) \) \( g(x)=\frac{1}{x} \) \( g(x)=x \)

To find the slope of the tangent line to the function \( g(x) = \frac{1}{x} \) at a specific point, we'll follow these steps: ### Given: - Function: \( g(x) = \frac{1}{x} \) - Point: \( (1, g(1)) \) ### Step 1: Find the Derivative of \( g(x) \) The derivative of a function gives us the slope of the tangent line at any point \( x \). \[ g'(x) = \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \] ### Step 2: Evaluate the Derivative at \( x = 1 \) Substitute \( x = 1 \) into the derivative to find the slope at that specific point. \[ g'(1) = -\frac{1}{(1)^2} = -1 \] ### Step 3: Conclusion The slope of the tangent line to the function \( g(x) = \frac{1}{x} \) at the point \( (1, 1) \) is **-1**. ### Additional Context (If Needed) If you're also considering the function \( g(x) = x \): - The derivative is \( g'(x) = 1 \), which means the slope of the tangent line at any point on this line is **1**. However, based on your query, the primary focus is on \( g(x) = \frac{1}{x} \), and the slope at \( (1, 1) \) is **-1**. **Answer:** For \( g(x) = \frac{1}{x} \) at the point (1, 1), the slope of the tangent line is –1.