Mathematics Departament ISV International School of Valencia BRITISH SCHOOL EL PLANTÍO Differentation Review Presentation Y12 Curriculum Maths Teachers: José Ramón Fierro, Head. Ignacio Muñoz Motilla.

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). Derivative of a function Derivative of f(x) = x 2 in x=2: To get the derivative in x=2: Derivative of f(x) = x 2 in x=3: f(x) = x 2 f ´(x) = 2x

f (x) f ' (x) Some examples of derivative functions Derivative of f(x) = k is f ' (x) = 0 Derivative of f(x) = x is f ' (x) = 1 f (x) f ' (x)

Cannot get the derivative Continuous function 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 Absolute value function f(x)=|x| is continuous in , but has no tangent line in x=0, thta means we cannot get the derivative.

Rules for differentiation y = a. f(x) y ' = a. f '(x) y = f(x)  g(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) y ' = f '(x). g(x) – f(x). g '(x) g 2 (x) y = f(x) (f –1 (x))' = 1 f '(y) / y = f(x)

The Chain Rule y = f [g(x)]y ' = f ' [g(x)].g'(x)  y ' = (sen ' t) (t )' = cos t. 2 = cos 2x. 2  y ' = (2t)' (t)' = 2. cos x 2x = t y = sen 2x y = 2 sen x sen x = t

DERIVATIVE OF RECIPROCAL CIRCULAR FUNCTIONS (I) 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 (II) X Y P(y, x) P ' (x, y) f f –1  90 –  f ' (y) = tg  (f –1 (x))' = tg (90 –  ) = 1 / tg  1 / f '(y) con f –1 (x) = y

DERIVATIVE OF LOGARITHMIC FUNCTIONS Let’s calculate the derivative of : } Using the reciprocal of a Function rule Sean Then the derivitaive of Will be;

Monotony: Growth and decrease in a range Regarding Average and Instantaneous rate of Change (ARC and IRC) [a[a ]b]b x f(x) x+h f(x+h) h Increasing function in [a, b] f(x) 0 ARC (x, h) > 0  (x, x+h) y h >0 [a[a ]b]b x h f(x) Decreasing function in [a, b] –ARC(x,h) f(x) 0 ARC (x, h) 0 f(x+h) x+h ARC(x,h)

STATIONARY POINTS f ' < 0 f ' > 0 f ' < 0 a b f ' (a) = 0 f " (a) > 0 f " (b) < 0 f ' (b) = 0 Local minimum of coordenates (a, f(a)) Local maximum of coordenates (b, f(b)) 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) ) 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.

Derivative of Sinus Function Let’s calculate the derivative of The derivative of will be Using the derivative definition

Derivative of the Tangent Function Let’s calculate the derivative of The derivative is Using the formula

Derivative of the arc sinus function Let’s calculate the derivative of } The derivative is As it is, Knowing Using the reciprocal of a Function rule

Derivative of the arc tangent function Let’s calculate the derivative of } The derivative will be As, it is Knowing that Using the reciprocal of a Function rule

y = f n (x) y '= n. f n–1 (x). f '(x) y = log a [f(x)] y ' = f '(x) f(x) · log a e y = a f(x) y ' = a f(x) · f '(x) · ln a y = sen f(x) More rules Function Its derivative function y ' = cos f(x). f '(x)y = cos f(x)y ' = – sen f(x). f '(x) y = tg f(x) y ' = f '(x) Cos 2 f(x)

y = arcsen f(x) y = arctg f(x) y = arccos f(x)y = arcctg f(x) More rules (II) Function Its derivative y ' = 1 + f 2 (x) – f '(x) y ' = 1 – f 2 (x) f '(x) y ' = 1 – f 2 (x) -f '(x) y ' = 1 + f 2 (x) f '(x)

Curvature: Convexity and Concavity [a[a ]b]b [a[a ]b]b [a[a ]b]b [a[a ]b]b increasing: Average Rate of Change positive and increasing: Convex function decreasing: Average Rate of Change negative and decreasing: Convex function increasing: Average Rate of Change positive and increasing: Concave function decreasing: Average Rate of Change negative and decreasing: Concave function

[a[a ]b]b Relations between the derivative function and curvature The gradients of the function increase  f ' is increasing  f " > 0  convex function [a[a ]b]b   x1x1 x2x2 tg    tg    f '(x 1 ) < f '(x 2 ) x1x1 x2x2  

[a[a ]b]b   [a[a ]b]b x1x1 x2x2   x1x1 x2x2 tg    tg    f '(x 1 ) > f '(x 2 ) The gradients of the function decrease  f ' is decreasing  f " < 0  Concave function Relations between the derivative function and curvature

Stationary Point of Inflection P(a, f(a)) f" < 0 f" > 0 f"(a) = 0

Summary regarding plotting a Graph of a Function 1. Study domain and continuity. 3. Intersection points with both axis 4. Get possible asymptotes. 5. Monotony. Study first derivative 6. Curvature. Get second derivative { { Vertical: Points that are not in the domain. Horizontals or obliquess: Getting limits in the infinity. 2. Check simetry and periodicity. { X-axis Y-axis: f (x) = 0 f (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 f “ (x) < 0

Plotting polynomial functions (I) Let’s plot the following function: R is its domain, it’s continuous and has no asymptotes 1. Interception points with both axis 2. Simetry 3. Limits in the infinity Y-axis: X-axis : { ODD

Sketching and Plotting the function 4. Monotony if Plotting polynomial functions (II)

Sketching and plotting 5. Curvature if Plotting polynomial functions (III)

Plotting Rational Functions (I) Let’s plot the following function 1. Domain and continuity 2. Interception points with axis 3. Simetry It has not Y axis: X axis:

6. Curvature if Plotting Rational Functions (II) The is not any stationary point of inflection Plotting and sketching the function