Tangent line at x=a has slope:

To find the slope of the tangent line at a specific point $x=a$ for a function $f(x)$, we use the concept of the derivative. 1. **Definition of the Derivative**: The derivative $f'(x)$ measures the instantaneous rate of change of the function $f(x)$ with respect to $x$. Geometrically, this is equivalent to the slope of the line tangent to the curve at any point $x$. 2. **Evaluating at a Point**: To find the slope at the specific point where $x=a$, we evaluate the derivative function at that value. This is denoted as $f'(a)$. 3. **Comparison with Options**: - $0$: This is only the slope if the tangent is horizontal (like at a local maximum or minimum). - $f(a)$: This is the $y$-coordinate (the value of the function), not the slope. - $f''(a)$: This is the second derivative, representing the curvature or concavity of the graph. - $f'(a)$: This is the first derivative, which represents the slope of the tangent line. Therefore, the slope of the tangent line at $x=a$ is $f'(a)$.