<p> Name______</p><p>The Definition of the Derivative</p><p>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. </p><p>(b) Approximate the slope of the tangent line by visual inspection (you are estimating the derivative of the curve by doing this).</p><p>(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)</p><p>(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: </p><p> f (x  h)  f (x) lim h0 h</p><p>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.</p><p>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).</p><p>Calculate the derivatives of the following functions by definition:</p><p>(1) f(x) = 3x2 +2x</p><p>(2) g(x) = 4/x</p><p>(3) h(x) = sin(x)</p><p>(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.</p>