Derivatives of some elementary functions presentation. Derivatives of some elementary functions

Rules of differentiation THEOREM 1. Differentiation of a sum, product and quotient. If the functions f and g are differentiable at a point x, then f + g, f g, f /g are differentiable at this point (if g(x) 0) and let y = f g. 1) (f(x) + g(x))" = f "(x) + g "(x); 2) (f(x) g(x))" = f "(x)g(x) + f(x)g "(x); Proof. Let us give a proof of property 2. f = f (x + x) – f(x) f (x + x) = f(x) + f ; g = g (x + x) – g(x) g(x + x)= g(x)+ g. g "(x) f "(x) 0 at x 0 (Due to the non-continuous differential function.)

THEOREM 2. Differentiation of a complex function Let the function y = f(u) be differentiable at the point u 0, y 0 = f(u 0), and the function u = (x) be differentiable at the point x 0, u 0 = (x 0). Then the complex function y = f ((x)) is differentiable at the point x 0 and f " ((x 0)) = f " (u 0)· " (x 0) or NOTE: The rule for calculating the derivative of a complex function extends to the composition of any finite number of functions. For example: (f ((g(x)))" = f "((g(x))) "(g(x)) g"(x). at point x and C = const, then (C f(x))" = C f "(x); (f(x)/C)" = f "(x)/C.

Example 1. y = cosx, x R. (cosx) = (sin(/2 – x)) = cos(/2 – x)·(/2 – x) = – sinx. y = tgx, x /2 + k, k Z. Using Theorems 1 and 2, we find the derivatives of the trigonometric functions y = ctgx, x + k, k Z.

THEOREM 3. Differentiation of the inverse function. If y = f(x) is continuous and strictly monotone on the interval and has a derivative f "(x 0), then its inverse function x = g(y) is differentiable at the point y 0 = f(x 0), and g "( y 0) = 1/ f "(x 0). x0x0 x 0 - x 0 + y0y0 x y x y y = f(x) x = g(y) Let y be such that y 0 + y (,). Let us denote x = g(y 0 + y) – g(y 0). We need to prove that 0 exists. Proof. Let f(x) strictly increase by . Let = f(x 0 -), = f(x 0 +). on [, ] an inverse function x = g(y) is defined, continuous and strictly increasing, and f(x 0) (,). If y 0, then x 0, due to the strict monotonicity of the function. Therefore, we have the right to write the identity If. y, then x, since x = g(y) is continuous at the point y 0.

Example 2. Find the derivatives of inverse trigonometric functions

0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2 " title=" Table of derivatives of elementary functions 1)(С)´= 0, C = const; 2)(x)´ = x -1, R, x > 0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2" class="link_thumb"> 8 !} Table of derivatives of elementary functions 1)(С)´= 0, C = const; 2)(x)´ = x -1, R, x > 0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2 x, x π/2 + πn, n; 8)(ctg x) = - 1/ sin 2 x, x πn, n; 9)10) 11)12) 0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2 "> 0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R ; (e x)´ = e x, x R; 5)(sin x) = cos x, x R; cos 2 x, x π/2 + πn, n ; 8)(ctg x) = - 1/ sin 2 x, x πn, n ; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2 " title=" Table of derivatives of elementary functions 1)(С)´= 0, C = const; 2)(x)´ = x -1, R, x > 0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2"> title="Table of derivatives of elementary functions 1)(С)´= 0, C = const; 2)(x)´ = x -1, R, x > 0; (x n)´ = n x n-1, n N, x R; 3)(a x)´ = a x lna, a > 0, a 1, x R; (e x)´ = e x, x R; 4). 5)(sin x) = cos x, x R; 6)(cos x) = - sin x, x R; 7)(tg x) = 1/ cos 2"> !}

Derivative of nth order DEFINITION. Let f(x) be defined in U (x 0) and have a derivative f (x) at each point in this interval. If at the point x 0 there is a derivative of f (x), then it is called the second derivative of the function f (x) at this point and is denoted. The derivative f (n) (x) of any order n = 1, 2, ... If in U (x 0) there is f (n-1) (x) (in this case, the zero-order derivative means the function itself), then n = 1, 2, 3, …. A function that has derivatives up to the nth order inclusive at each point of the set X is called n times differentiable on the set X.

Let the functions f(x) and g(x) have nth order derivatives at the point x. Then the function Аf(x) + Вg(x), where А and В are constant, also has a derivative at the point x, and (Аf(x) + Вg(x)) (n) = Аf (n) (x) + Вg (n)(x). When calculating derivatives of any order, the following basic formulas are often used. y = x ; y (n) = (-1)... (- (n-1)) x - n. y = x -1, y = (-1)x -2, y = (-1)(-2) x -3 ... In particular, if = m N, then y = a x ; y (n) = a x (lna) n. y = a x lna, y = a x (lna) 2, y = a x (lna) 3, ... In particular (e x) (n) = e x. y " = ((x + a) - 1)" = - (x+a) - 2, y "" = 2 (x + a) - 3, y """ = (x + a) - 4, …

Y = ln(x+a); y (n) = (–1) n–1 (n–1)!(x+a) –n. y = (x +a) –1, y = – (x +a) –2, y = 2(x +a) –3, y (4) = – 2 3(x +a) – 4, … y = sin αx; y (n) = α n sin(αx+n· /2) y = α cos αx = α sin(αx+ /2), y = α 2 cos(αx+ /2) = α 2 sin(αx+2· / 2), y = α 3 cos(αx + 2· /2) = α 3 sin(αx+3· /2), ... y = cos αx; y (n) = α n cos(αx+n· /2) y = – α sin αx = α cos(αx+ /2), y = – α 2 sin(αx+ /2) = α 2 cos(αx + 2 · /2), y = – α 3 sin(αx+2· /2) = α 3 cos(αx + 3· /2),...

The Nth derivative of the product of two functions (Leibniz formula) where This formula is called the Leibniz formula. It can be written in the form where Let the functions f(x) and g(x) have nth order derivatives at point x. By induction we can prove that (f(x) g(x)) (n) = ?
Example 5. y = (x 2 +3x+5) sin x, y (13) = ? = sin(x +13π /2) (x 2 +3x+5) + 13 sin (x +12π /2) (2x+3) + 78 sin (x +11π /2) 2 = = cos x (x 2 +3x+5) + 13 sin x (2x+3) + 78 (- cos x) 2 = = (x 2 +3x -151) cos x + 13 (2x+3) sin x. Let us apply the Leibniz formula, putting in it f(x) = sin x, g(x) = (x 2 +3x+5). Then

Slide 1

Derivative of a function Definition of a derivative Geometric meaning of a derivative Relationship between continuity and differentiability Derivatives of basic elementary functions Rules of differentiation Derivative of a complex function Derivative of an implicit function Logarithmic differentiation

Slide 2

Definition of derivative Let the function y = f(x) be defined in some interval (a; b). Let's give the argument x some increment: x f(x) x+Δx f(x+ Δx) Let's find the corresponding increment of the function: If there is a limit, then it is called the derivative of the function y = f(x) and is denoted by one of the symbols:

Slide 3

Definition of derivative So, by definition: A function y = f(x) that has a derivative at each point of the interval (a; b) is called differentiable in this interval; the operation of finding the derivative of a function is called differentiation. The value of the derivative of the function y = f(x) at the point x0 is denoted by one of the symbols: If the function y = f(x) describes any physical process, then f ’(x) is the speed of this process - the physical meaning of the derivative.

Slide 4

Geometric meaning of the derivative Let's take two points M and M1 on a continuous curve L: x f(x) x+Δx M M1 f(x+ Δx) We draw a secant through the points M and M1 and denote by φ the angle of inclination of the secant.

Slide 5

Geometric meaning of the derivative The derivative f ’(x) is equal to the slope of the tangent to the graph of the function y = f(x) at the point whose abscissa is x. If the tangent point M has coordinates (x0; y0), the slope of the tangent is k = f ’(x0). Equation of a line with slope: A line perpendicular to the tangent at the point of tangency is called the normal to the curve. Tangent equation Normal equation

Slide 6

Relationship between continuity and differentiability of a function If a function f(x) is differentiable at a certain point, then it is continuous at that point. Theorem Let the function y = f(x) be differentiable at some point x, therefore there is a limit: Proof: where at According to the theorem on the connection between a function, its limit and an infinitesimal function, the function y = f(x) is continuous. The converse is not true: a continuous function may not have a derivative.

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Derivatives of basic elementary functions 1 Newton’s binomial formula: Power function: K – factorial

Slide 8

Derivatives of the main elementary functions According to the Newton binomial formula we have: Then:

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Derivatives of basic elementary functions 2 Logarithmic function: Rules for differentiation of other basic elementary functions are derived similarly.

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Rules of differentiation Let u(x), v(x) and w(x) be functions differentiable in a certain interval (a; b), C is a constant.

Slide 11

Derivative of a complex function Let y = f(u) and u = φ(x), then y = f(φ(x)) is a complex function with an intermediate argument u and an independent argument x. Theorem This rule remains in force if there are several intermediate arguments:

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Slide 13

DERIVATIVE

Municipal educational institution Srednesantimirskaya secondary school

Completed by a math teacher

Singatullova G.Sh.

Definition of derivative.
Physical meaning of derivative.
.

Basic rules of differentiation.
Derivative of a complex function.
Examples of solving problems on the topic derivative.

Definition of derivative

Let the function y= be defined on some interval (a, b) f(x). Let's take any point x 0 from this interval and give the argument x at the point x 0 an arbitrary increment ∆ x such that the point x 0 + ∆ x belongs to this interval. The function will be incremented

Derivative functions y= f(x) at the point x =x 0 is called the limit of the ratio of the increment of the function ∆y at this point to the increment of the argument ∆x, when the increment of the argument tends to zero.

Geometric meaning of derivative

Let function y= f(x) is defined on some interval (a, b). Then the tangent of the angle of inclination of the secant MR to the graph of the function.

Where  is the angle of inclination of the tangent function f(x) at point (x 0 , f(x 0)).

The angle between curves can be defined as the angle between the tangents drawn to these curves at any point.

Equation of a tangent to a curve:

Physical meaning of the derivative 1. The problem of determining the speed of movement of a material particle

Let a point move along a certain line according to the law s= s(t), where s is the distance traveled, t is the time, and it is necessary to find the speed of the point at the moment t 0 .

By the moment of time t 0, the distance traveled is equal to s 0 = s(t 0), and by the moment (t 0 + ∆t) - the path s 0 + ∆s=s(t 0 + ∆t).

Then over the interval ∆t the average speed will be

The smaller ∆t, the better the average speed characterizes the movement of a point at the moment t 0. Therefore, under speed of the point at time t 0 should be understood as the limit of the average speed for the period from t 0 to t 0 +∆t, when ∆t⇾0, i.e.

2. PROBLEM ABOUT THE RATE OF CHEMICAL REACTIONS

Let a substance undergo a chemical reaction. The amount of this substance Q changes during the reaction depending on time t and is a function of time. Let the amount of substance change by ∆Q during time ∆t, then the ratio will express the average rate of a chemical reaction during time ∆t, and the limit of this ratio

Current rate of chemical reaction

time t.

3. TASK DETERMINATIONS OF RADIOACTIVE DECAY RATE

If m is the mass of a radioactive substance and t is time, then the phenomenon of radioactive decay at time t, provided that the mass of the radioactive substance decreases over time, is characterized by the function m = m(t).

The average decay rate over time ∆t is expressed by the ratio

and the instantaneous decay rate at time t

ALGORITHM for calculating the derivative

The derivative of the function y= f(x) can be found using the following scheme:

1. Let's give the argument x an increment ∆x≠0 and find the incremented value of the function y+∆y= f(x+∆x).

2. Find the increment of the function ∆y= f(x+∆x) - f(x).

3. Create a relationship

4. Find the limit of this ratio at ∆x⇾0, i.e.

(if this limit exists).

Basic rules of differentiation

Let u=u(x) And v=v(x) – differentiable functions at point x.

1) (u  v)  = u   v 

2) (uv)  = u  v +uv 

(cu)  =cu 

3) , If v  0

Derivative of a complex function

Theorem. If a function is differentiable at a point x, and the function

is differentiable at the corresponding point, then the complex function is differentiable at the point x, and:

those. the derivative of a complex function is equal to the product of the derivative of the function with respect to the intermediate argument and the derivative of the intermediate argument with respect to x.

Task 1.

Problem 2 .

Problem 3 .

Problem 4 .

Problem 5 .

Problem 6 .

Problem 7 .

Problem 8 .

Derivatives of some elementary functions presentation. Derivatives of some elementary functions

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