Definition of Derivative
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1 Definition of Derivative The derivative of the function f with respect to the variable x is the function ( ) fʹ x whose value at xis ( x) fʹ = lim provided the limit exists. h 0 ( + ) ( ) f x h f x h Slide 3.1-1
2 Derivative at a Point (alternate) The derivative of the function f at the point where x= a is the limit ( a) fʹ = lim provided the limit exists. x a ( ) ( ) f x f a x a Slide 3.1-2
3 Notation There are many ways to denote the derivative of a function y = f( x). Besides f '( x), the most common notations are: yʹ y prime Nice and brief, but does not name the independent variable. dy dy dx or the derivative Names both variables and dx of y with respect to x uses d for derivative. df dx df dx or the derivative of f with respect to x Emphasizes the function s name. d f ( x ) d dx of f at x or the Emphasis that differentiation dx derivative of f at x is an operation performed on f. Slide 3.1-3
4 One-sided Derivatives ( ) differentiable on a closed interval [ a, b] A function y= f x is if it has a derivative at every interior point on the interval, and if the limits ( + ) ( ) f a h f a lim+ h 0 h f ( b+ h) f ( b) lim h 0 h exist at the endpoints. In the right-hand derivative, [ the right - hand derivative at a] [ the left - hand derivative at b] h is positive and a+ h approaches a from the right. In the left-hand derivative, h is negative and b+ approaches b from the left. h Slide 3.1-4
5 Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will straighten out when we zoom in on them at a point of differentiability. Slide 3.2-5
6 Differentiability Implies Continuity If f has a derivative at x= a, then f is continuous at x= a. The converse of Theorem 1 is false. A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point. Slide 3.2-6
7 Intermediate Value Theorem for Derivatives Not every function can be a derivative. If a and b are any two points in an interval on which f is differentiable, then ( ) ʹ( ) fʹ a and f b. f ʹ takes on every value between Slide 3.2-7
8 Rule 1 Derivative of a Constant Function If f is the function with the constant value c, then df d = ( c ) = 0 dx dx This means that the derivative of every constant function is the zero function. Slide 3.3-8
9 Rule 2 Power Rule for Positive Integer Powers of x. If n is a positive integer, then d x nx dx The Power Rule says: n n 1 ( ) = n To differentiate x, multiply by n and subtract 1 from the exponent. Slide 3.3-9
10 Rule 3 The Constant Multiple Rule If u is a differentiable function of x and c is a constant, then d du ( cu) = c dx dx This says that if a differentiable function is multiplied by a constant, then its derivative is multiplied by the same constant. Slide
11 Rule 4 The Sum and Difference Rule If u and v are differentiable functions of x, then their sum and differences are differentiable at every point where u and v are differentiable. At such points, d du dv ( u± v) = ± dx dx dx. Slide
12 Rule 5 The Product Rule The product of two differentiable functions u and v is differentiable, and d dv du ( uv) = u + v dx dx dx The derivative of a product is actually the sum of two products. Slide
13 Rule 6 The Quotient Rule u At a point where v 0, the quotient y= of two differentiable v functions is differentiable, and du dv v u d u = dx dx 2 dx v v Since order is important in subtraction, be sure to set up the numerator of the Quotient rule correctly. Slide
14 Rule 7 Power Rule for Negative Integer Powers of x If n is a negative integer and x 0, then d 1 ( x n) nx n =. dx This is basically the same as Rule 2 except now n is negative. Slide
15 Instantaneous Rates of Change ( ) The instantaneous rate of changeof is the derivative fʹ provided the limit exists. ( a) = lim h 0 f with respect to x at a ( + ) ( ) f a h f a When we say rate of change, we mean instantaneous rate of change. h Slide
16 Instantaneous Velocity ( ) = ( ) The instantaneous velocity is the derivative of the position function s f t with respect to time. At time t the velocity is ( ) vt ds = = dt Δ t 0 ( +Δ ) ( ) f t t f t lim. Δt Slide
17 Speed Speed is the absolute value of velocity. Speed ( ) = vt = ds dt Slide
18 Acceleration Acceleration is the derivative of velocity with respect to time. ds If a body s velocity at time t is v( t) = then the body s dt 2 dv d s acceleration at time t is a( t) = =. 2 dt dt Slide
19 Free-fall Constants (Earth) ft English Units g= 32, s= 2 ( 32) t = 16t sec 2 m 1 Metric Units g= 9.8, s= 2 ( 9.8) t = 4.9t sec Slide
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