The Definition of the Derivative

Name______

We will find the derivative of a function at a point from scratch. Consider the function f(x) = x2. We want to find the derivative of f(x) at the point (2, 4). (a) First graph the function, and draw a rough tangent line to the graph.

(b) Approximate the slope of the tangent line by visual inspection (you are estimating the derivative of the curve by doing this).

(c) Now find the slope of the secant lines through (2, 4) and (3,9); (2, 4) and (2.5, (2.5)2); (2, 4) and (2.1, (2.1)2); (2, 4) and (2.01, (2.01)2); (2, 4) and (2.001, (2.001)2); (2, 4) and (2 + x, (2 + x)2) (x represents a small change in x)

(d) Find the slope of the tangent line by taking the limit of the last slope in (c) as x  0; recall the slope of the tangent line is the derivative of the function at that point, so you have now found the derivative f ' (2). So our definition of the derivative, f ' (x), of the function f(x) is derived from taking the limit of slopes of secant lines, and is given by:

f (x  h)  f (x) lim h0 h

If this limit exists at a point, then we say f is differentiable at that point. If the limit does not exist at the point, then f is not differentiable at the point. A function that is differentiable everywhere has a derivative for all real numbers.

Fact: If a function is differentiable at a point then it is continuous at that point. (So if a function isn’t continuous at a point then it isn't differentiable at that point).

Calculate the derivatives of the following functions by definition:

(1) f(x) = 3x2 +2x

(2) g(x) = 4/x

(3) h(x) = sin(x)

(4) k(x) = |x|. Note for a piecewise function, if you aren't at a point where the definition changes, then you only use one piece of the function. What I am saying is that there are 3 cases to consider here. x< 0 (then k(x) = -x), x > 0 (where k(x) = x), and then the point in the middle x=0 when you have a left- and right- limit for the derivative just as with checking continuity. This case shows that a function can be continuous at a point yet not differentiable at that point.