Sections 2.1/2.2 The Derivative

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1 1 Sections 2.1/2.2 The Derivative In developing the idea of the it we talked about the derivative (recall sections 1.1/1.2) as 1) a rate of change or instantaneous velocity 2) the slope of the tangent line. We now redefine it in terms of the its without ɛ s and δ s. Definition The derivative of a function f at a point a, denoted by f (a) is f (a) x a if this it exists and we say the function f is differentiable at x = a. Remarks 1) This definition reinforces the geometric concept of the derivative Slope of tangent line to f(x) at (a, f(a)) = f (a) x a it of slopes of secant lines through (a, f(a)) and (x, f(x)) 2) Other than the one we have seen in the definition there is a different but equivalent way to evaluate the slope of the tangent line or derivative of a function f(x) at a point x=a. Let h = x a represent the distance between x and a. Since this gives x = a + h you can substitute in the previous expression and get Slope of tangent line to f(x) at (a, f(a)) = f (a) f(a + h) f(a) h }{{} called difference quotient Both of these expressions are equivalent ways to compute the derivative, and you should feel comfortable using at least one of them.

2 2 Example Find the derivative of the function f(x) = x 2 8x + 9 at the number a. x 2 8x + 9 (a 2 8a + 9) x a x a x 2 a 2 8x + 8a x a (x a)(x + a) 8(x a) x a (x + a 8) = a + a 8 = 2a 8 Since the it exists for any value of a, f (a) = 2a 8. Example Use the second definition given in Remarks (2) to calculate f (a) for f(x) = x 2 8x + 9. f f(a + h) f(a) (a + h) 2 8(a + h) + 9 [a 2 8a + 9] (a) h h a 2 + 2ah + h 2 8a 8h + 9 a 2 + 8a 9 h 2ah 8h + h 2 h (2a 8 + h) = 2a 8 Note the point a in the last two examples was arbitrary so we can declare that we found the derivative of f(x) at all points in its domain f(x) = x 2 8x + 9 and f (x) = 2x 8 Example Let f(x) = x 2 8x + 9. Find the equation of the tangent line to the curve y = f(x) at the point (3, 6). Note that f(3) = 3 2 8(3) + 9 = 6. We also know that f (3) = slope of the tangent line at (3, f(3)). f (3) = 2(3) 8 = 2. So the equation of the tangent line through (3, 6) with slope m = 2 is: y ( 6) = ( 2)(x 3) y = 2x

3 3 You can observe this in the figure below Out[23]= Definition The derivative of f(x) is the function f (x) given by f (x) f(x + h) f(x) h provided the it exists. (So domain of f (x) is those x s where the above it exists.) Remarks: 1) The process of computing a derivative is called differentiation. 2) f is differentiable on an interval I if it is differentiable at every point in I.

4 4 Example The position of a particle is given by the equation of motion s(t) = 1 t+1 where t is measured in seconds and s in meters. a) Find the velocity function s (x) b) Find the velocity of the particle at t = 2, 5 and t = 10 seconds. a) b) s s(t + h) s(t) (t) h 1 1+(t+h) 1 1+t h 1 + t (1 + t + h) (1 + t + h)(1 + t) h h(1 + t + h)(1 + t) (1 + t + h)(1 + t) = (1 + t) 2 s (2) = s (5) = s (10) = (1 + 2) 2 = 9 (1 + 5) 2 = 36 (1 + 10) 2 = 121 Theorem If f is differentiable at a, then f is continuous at a. Proof To prove continuity, we need to show f(x) = f(a). This is equivalent to show that = 0. But since f(a) does not depend on x, we can push it inside the it () = 0 So if we show we will prove the continuity. a () ()x ( x a ) (x a) ( ) x a

5 Since f(x) f(a) x a break up (*) above by the Limit Laws = f (a) (i.e. the its exists) and x a = 0, we can () (x a) x a = f (a) 0 = 0 Cautionary Remark Above theorem says differentiability continuity. But the converse is not true continuity differentiability. An example of this is f(x) = x which is not differentiable at x=0 but it is continuous at zero.(more on this below) Although there are many ways a function could fail to be differentiable at a point a, there are three typical types of non-differentiability. (1)Discontinuities A function is not differentiable at a point where the graph of f is not continuous. This is a direct result of the above theorem and the remark following it. (2)Corner A function is not differentiable at a point where the graph of f has a kink or corner (one might also call such a point on the graph a kink). Essentially, these places fail to be differentiable because the left and right hand its x a and + x a do not match up. For instance, the absolute value function f(x) = x fails to be differentiable at 0 because f(x) f(0) x 0 = x x 0 x 0 x 0 x x = x x = 5 and f(x) f(0) x 0 = x x 0 + x 0 x 0 + x x = x x = 1 A special case of a corner point non-differentiability is called a cusp where one of the one-sided it above is and the other one-sided it is. The picture below is an example of a cusp.

6 6 Out[16]= Vertical Tangents Finally, a function is not differentiable at a point on the graph where the tangent line to f is a vertical line. This is because the slope of the tangent to the graph at this point is infinite, which is also in your book corresponds to does not exist. Example The following function displays all 3 failures of differentiability a corner (at x=-1), discontinuity (at x=0) and a vertical tangent (at x=1). ( m=0 is the slope of the tangent lines when x < 2, m=-1 is the slope of the tangent lines when 2 < x < etc. except m = + is the slope of the tangent line at x=1(when approaching from right.))

7 7 Below you have the graph of y = f (x) Alternative Derivative Notations Let f(x) be a function the following are possible notations for the derivative of f: f (x) y says take the derivative of whatever fol- Leibniz Notation: Operator Notation : lows). dy dx = df dx d dx (f(x)) ( d dx