How to find the slope of a tangent to a curve at a specific point using a given function?

To find the slope of a tangent to a curve at a specific point using a given function, you'll use the concept of derivatives. Here's a step-by-step process:

1. Find the derivative of the given function: The derivative of a function at a specific point gives you the slope of the tangent line at that point. If you have a function $f(x)$, find its derivative $f'(x)$.

2. Evaluate the derivative at the specific point: Once you have the derivative, substitute the x-value of the specific point where you want to find the slope. Let's call this point $x=a$. So, evaluate $f'(a)$.

3. Write the equation of the tangent line: Now that you have the slope of the tangent line ($f'(a)$), you can write the equation of the tangent line using the point-slope form:

   $$y - f(a) = f'(a)(x - a)$$

Here's an example to illustrate the process:

Given the function $f(x) = x^2$, find the slope of the tangent line at $x=3$ and write the equation of the tangent line.

1. Find the derivative: $f'(x) = 2x$

2. Evaluate the derivative at $x=3$: $f'(3) = 2 \cdot 3 = 6$

3. Write the equation of the tangent line using the point $(3, f(3)) = (3, 9)$:

   $$y - 9 = 6(x - 3)$$