Using the Point-Slope Form in Calculus

In algebra you learned to use the point-slope form to find the equation of a line going crossing a specific point (x, y) with a specific slope.

In this equation, (x₁, y₁) are the coordinates of the point and m is the slope of the line. Explore the point-slope form in an interactive Desmos graphing window. 

What does this have to do with calculus?

Suppose there is differentiable function f(x). We know that f’(x) is another function that can be used to find the slope of the line tangent to f(x) at a given point.

In this graph, the blue line is tangent to the curve at a specific point. How can we find the equation of that line?

This is where the calculus version of the point-slope form comes in.

Look at this graph. 

The graph of the function f(x) is shown in red. The graph of the line tangent to f(x) crosses the coordinates (a, f(a)), for some value a. 

We know that f’(x) is the function that you can use to find the slope of the line at a specific point.

What does f’(a) represent? Since f’(x) is a function for generating slopes, that means that f’(a) is the slope of the line tangent to f(x) that crosses (a, f(a)).

We now have all that we need to find the equation of the line using the point slope form. This is the calculus version of the point-slope form:

The only difference between this version and the point-slope form from algebra is the f’(a) term. But since this is the slope of the line at (a, f(a)), then f’(a) is the same as the term m in the algebra version.

Explore the calculus version of the point-slope form in an interactive Desmos graphing window.