The derivative of f(x) at point a is the slope of the tangent line at point a or . Allow me to explain.



As seen in the picture above, a secant line is a line that passes through two points of a function and a tangent line touches only one. The function to find the slope of a secant line that passes through is where h is the change in x or . This is simply the slope formula, , modified. Taking the limit as h approaches 0 means that the would “become zero”, so the line would only pass through that one point. That is a tangent line.

The picture above should help make it clearer. It illustrates using the derivative formula to find the slope of the tangent line at x=1 on . The red line is the secant line as h approaches zero from the left (values of h are -3, -2, and -1). The blue line is the secant line as h approaches zero from the right (values of h are 3 and 2). The purple line is the tangent line; it touches the function at only one point. Now I will show you how it is done algebraically. I have written it out since I believe that is easier to understand.

The first thing I do is replace f(x) with its value replacing x with x + h in the first term. I then multiply everything out so I am left with a polynomial. The ‘s and 5‘s cancel out. Now it is possible to cancel out an h in each term. After doing so, I plugged in zero for h to leave me with the derivative -2x.

The derivative only tells you the slope of the tangent line, not the equation. Since you know the slope of the line and a point on the line, it is not difficult to find the equation of the tangent line, g(x) in this case.

There are many tricks and shortcuts to finding the derivative of a line. The derivative formula is usually the most difficult. Things to remember: The derivative of is . A derivative of the derivative yields the second derivative or . The derivative of y is written as . In the next two posts I will delve into some of the shortcuts to finding the derivative of a function and why derivatives are useful.