Derivative of a function

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

Derivative of a function