132 Views

Download Presentation
## Derivative of a function

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Mathematics DepartamentISV International School of**ValenciaBRITISH SCHOOL EL PLANTÍODifferentation Review PresentationY12 CurriculumMaths Teachers:José Ramón Fierro, Head.Ignacio Muñoz Motilla.**Derivative of f(x) = x2 in x=3:**Derivative of a function f(x) = x2 • Derivative of f(x) = x2 in x=2: f ´(x) = 2x • To get the derivative in x=2: • Notice that the expression f ´(x) is itself a function and for this reason we also refer to the derivative as the gradient function of y = f(x).**f (x)**f ' (x) f (x) f ' (x) Derivative of f(x) = k is f ' (x) = 0 Some examples of derivative functions Derivative of f(x) = x is f ' (x) = 1**If a function has a derivative in one point “P”, if**exists the straight line tangent (not vertical) in this point to the graph, that means that the graph in this point is continuous. Derivative function (Differentation) and continuity Continuous function • Absolute value function f(x)=|x| is continuous in , but has no tangent line in x=0, thta means we cannot get the derivative. Cannot get the derivative**f(x)**f '(x) . g(x) – f(x) . g '(x) y = 1 g(x) y ' = / y = f(x) (f–1(x))' = g2(x) f '(y) y ' = a . f '(x) y = a . f(x) y ' = f '(x) g '(x) y = f(x) g(x) y ' = f '(x) . g(x) + g '(x) . f(x) y = f(x) . g(x) Rules for differentiation y = f(x)**y = f [g(x)]**y ' = f ' [g(x)] .g'(x) 2x = t sen x = t y ' = (sen ' t) (t )' = cos t . 2 = cos 2x . 2 y = sen 2x The Chain Rule y = 2 sen x y ' = (2t)' (t)' = 2. cos x**Let’s get the derivative of function**} 2. With the chain rule (f o g)’ (x) = f ’ (g(x)) g’(x) DERIVATIVE OF RECIPROCAL CIRCULAR FUNCTIONS (I)**Y**90 – a a X f–1 f P ' (x, y) • f ' (y) = tg a P(y, x) DERIVATIVE OF RECIPROCAL CIRCULAR FUNCTIONS (II) • (f –1(x))' = tg (90 – a) = 1 / tg a = • 1 / f '(y) con f –1(x) = y**Let’s calculate the derivative of :**} Sean Using the reciprocal of a Function rule DERIVATIVE OF LOGARITHMIC FUNCTIONS Then the derivitaive of Will be;**ARC(x,h)**–ARC(x,h) h f(x+h) h ] b ] b [ a [ a x x+h x x+h f(x) f(x+h) f(x) Monotony: Growth and decrease in a range Regarding Average and Instantaneous rate of Change (ARC and IRC) Increasing function in [a, b] Decreasing function in [a, b] f(x) < f(x+h), (x, x+h) y h >0 f(x) < f(x+h), (x, x+h) y h >0 ARC (x, h) > 0 (x, x+h) y h >0 ARC (x, h) < 0 (x, x+h) y h >0**f " (b) < 0**f ' (b) = 0 a f ' < 0 f ' > 0 f ' < 0 b f ' (a) = 0 f " (a) > 0 Local maximum of coordenates (b, f(b)) STATIONARY POINTS Local minimum of coordenates (a, f(a)) So far we have discussed the conditions for a function to be increasing ( f '(x) > 0) and for a function to be decreasing ( f '(x) < 0). What happens at the point where a function changes from an increasing state (( f '(x) > 0) ) to ( f '(x) = 0) and then to a decreasing state (( f '(x) < 0) ) or vice–versa? Points where this happens are known as stationary points. At the point where the function is in a state where it is neither increasing nor decreasing, we have that f '(x) = 0 . There are times when we can call these stationary points stationary points, but on such occassions, we prefer the terms local maximum and local minimum points.**Let’s calculate the derivative of**The derivative of will be Using the derivative definition Derivative of Sinus Function**Let’s calculate the derivative of**Using the formula The derivative is Derivative of the Tangent Function**Let’s calculate the derivative of**} Knowing The derivative is As it is, Using the reciprocal of a Function rule Derivative of the arc sinus function**Let’s calculate the derivative of**} Knowing that The derivative will be As, it is Using the reciprocal of a Function rule Derivative of the arc tangent function**Its derivative function**y = f n(x) Function y '= n . f n–1(x) . f '(x) f '(x) y ' = y = loga[f(x)] Cos2 f(x) f '(x) · loga e y ' = y = af(x) f(x) y ' = cos f(x) . f '(x) y ' = – sen f(x) . f '(x) y ' = af(x) ·f '(x) · ln a y = cos f(x) y = tg f(x) y = sen f(x) More rules**Its derivative**Function f '(x) y ' = 1 – f2(x) -f '(x) y ' = y = arccos f(x) 1 – f2(x) f '(x) y ' = y = arcctg f(x) 1 + f2(x) – f '(x) y ' = 1 + f2(x) y = arcsen f(x) More rules (II) y = arctg f(x)**]**b [ a ] b ] b ] b [ a [ a [ a AverageRate of Change positive and increasing:Convexfunction AverageRate of Changenegative and decreasing:Convexfunction Curvature: Convexity and Concavity AverageRate of Changenegative and decreasing:Concavefunction AverageRate of Change positive and increasing:Concavefunction**a1**a2 a2 a1 [ a [ a ] b ] b x1 x2 x1 x2 Relations between the derivative function and curvature tg a1 < tg a2 f '(x1) < f '(x2) The gradients of the function increase f ' is increasing f " > 0 convex function**a2**a1 a2 a1 ] b [ a x1 x2 x1 x2 ] b [ a Relations between the derivative function and curvature tg a1 > tg a2 f '(x1) > f '(x2) The gradients of the function decrease f ' is decreasing f " < 0 Concave function**f" < 0**f" > 0 P(a, f(a)) Stationary Point of Inflection f"(a) = 0**{**X-axis Y-axis: f (x)= 0 { f (0) Vertical: Points that are not in the domain. Horizontals or obliquess: Getting limits in the infinity. { f ‘ (x)= 0 Posible stationary: Growth: Decreasing: f ‘ (x)> 0 f ‘ (x) < 0 f “ (x)= 0 Posible Inflection Points: Convex: Concave: f “ (x)> 0 f “ (x)< 0 1. Study domain and continuity. 2. Check simetry andperiodicity. 3. Intersection points with both axis 4. Get possible asymptotes. Summary regarding plotting a Graph of a Function 5. Monotony. Study first derivative { 6. Curvature. Get second derivative**Let’s plot the following function:**Y-axis: { X-axis : R is its domain, it’s continuous and has no asymptotes 1. Interception points with both axis Plotting polynomial functions (I) 2. Simetry ODD 3. Limits in the infinity**Sketching and Plotting the function**if if if 4. Monotony Plotting polynomial functions (II)**Sketching and plotting**if if Plotting polynomial functions (III) 5. Curvature**Let’s plot the following function**Y axis: X axis: 1. Domain and continuity 2. Interception points with axis Plotting Rational Functions (I) 3. Simetry It has not**Plotting and sketchingthefunction**if if 6. Curvature Plotting Rational Functions (II) The is not any stationary point of inflection