Examples of calculation of derivative

See Derived for more total information.

The derived is a mathematical function, more precisely a function of functions because it takes as argument of entry a function and returns another function, generally different.

Examples starting from the definition of the derived number

Constant function

That is to say C a reality.

Let us consider the constant function F of value C :

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash \; forall\; H\; \backslash \; in\; \backslash \; mathbb\; \{R^*\},\; \backslash \; frac\; \{F\; (x+h)\; -\; F\; (X)\}\{H\}\; =\; \backslash \; frac\; \{DC\}\; \{H\}\; =\; 0$

thus

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; f\text{'}\; (X)\; =\; \backslash \; lim\_\; \{H\; \backslash \; rightarrow\; 0\}\; \backslash \; frac\; \{F\; (x+h)\; -\; F\; (X)\}\{H\}\; =0$.

Thus the derivative of a constant function is the null function.

Function square

Let us consider the function F definite on $\backslash \; mathbb\; \{R\}$ by

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; F\; (X)\; =x^2$

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; \backslash \; forall\; H\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; ^*,\; \backslash \; frac\; \{F\; (x+h)\; -\; F\; (X)\}\{H\}\; =\; \backslash \; frac\; \{(x+h)\; ^2-x^2\}\; \{H\}$

$=\; \backslash \; frac\; \{(x+h-x)\; (x+h+x)\}\{H\}\; =\; \backslash \; frac\; \{H\; (2x+h)\}\{H\}\; =2x+h$

thus

$f\text{'}\; (X)\; =\; \backslash \; lim\_\; \{H\; \backslash \; rightarrow\; 0\}\; (2x\; +\; H)\; =2x$

the derivative of F is thus the function F ' defined by

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\},\; f\text{'}\; (X)\; =2x$.

Function root

Let us consider the function F =√x

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; \_+^*,\; \backslash \; forall\; H\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; ^*,\; h>-x,\; \backslash \; quad\; \backslash \; frac\; \{F\; (x+h)\; -\; F\; (X)\}\{H\}\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{x+h\}\; -\; \backslash \; sqrt\; \{X\}\}\; \{H\}$

$=\; \backslash \; frac\; \{(\backslash \; sqrt\; \{x+h\}\; -\; \backslash \; sqrt\; \{X\})\; (\backslash \; sqrt\; \{x+h\}\; +\; \backslash \; sqrt\; \{X\})\}\{H\; (\backslash \; sqrt\; \{x+h\}\; +\; \backslash \; sqrt\; \{X\})\}$

$=\; \backslash \; frac\; \{x+h\; -\; X\}\; \{H\; (\backslash \; sqrt\; \{x+h\}\; +\; \backslash \; sqrt\; \{X\})\}=\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{x+h\}\; +\; \backslash \; sqrt\; \{X\}\}$

thus

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; \_+^*,\; f\text{'}\; (X)\; =\; \backslash \; lim\_\; \{H\; \backslash \; rightarrow\; 0\}\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{x+h\}\; +\; \backslash \; sqrt\; \{X\}\}\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; sqrt\; \{X\}\}$

In addition,

$\backslash \; forall\; H\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; \_+^*,\; \backslash \; frac\; \{F\; (H)\; -\; F\; (0)\}\{H\}\; =\; \backslash \; frac\; \{\backslash \; sqrt\; \{H\}\}\; \{H\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{H\}\}$

$\backslash \; lim\_\; \{H\; \backslash \; rightarrow\; 0\}\; \backslash \; frac\; \{F\; (H)\; -\; F\; (0)\}\{H\}\; =+\; \backslash \; infty$

thus F is not derivable into 0 and the representative curve admits into 0 a half vertical tangent.

Examples starting from the formulas of derivative

Here a series of examples of derivative calculated starting from the formulas established by the method with the limit.

Second degree

Let us consider the following functions and then derive them thereafter:

1. $y=x^2+5x-3\; \backslash ,$

2. $y=3x^2-9x+\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash ,$

3. $y=-4x^2+\; \backslash \; frac\; \{4\}\; \{7\}\; x-1\; \backslash ,$

Derived: 1. $y=x^2+5x-3\; \backslash ,$

$y\text{'}=\; (x^2)\; “+\; (5x)”\; -\; (3)\; \text{'}\backslash ,$

$y\text{'}=2x+5+0\; \backslash ,$

$y\text{'}=2x+5\; \backslash ,$

2. $y=3x^2-9x+\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash ,$

$y\text{'}=\; (3x^2)\; \text{'}-\; (9x)\; “+\; (\backslash \; frac\; \{2\}\; \{3\})”\; \backslash ,$

$y\text{'}=6x-9+0\; \backslash ,$

$y\text{'}=6x-9\; \backslash ,$

3. $y=-4x^2+\; \backslash \; frac\; \{4\}\; \{7\}\; x-1\; \backslash ,$

$y\text{'}=\; (-\; 4x^2)\; “+\; (\backslash \; frac\; \{4\}\; \{7\}\; X)”\; -\; (1)\; \text{'}\backslash ,$

$y\text{'}=-8x+\; \backslash \; frac\; \{4\}\; \{7\}\; -\; 0\; \backslash ,$

$y\text{'}=-8x+\; \backslash \; frac\; \{4\}\; \{7\}\; \backslash ,$

Third degree

Let us consider the following functions and derive them thereafter:

1. $y=2x^3+6x^2-4x+\; \backslash \; frac\; \{9\}\; \{\backslash \; pi\}\; \backslash ,$

2. $y=-x^3-5x^2+\; \backslash \; frac\; \{2\}\; \{3\}\; x-1\; \backslash ,$

3. $y=\; \backslash \; frac\; \{5\}\; \{17\}\; x^3+x^2-2x+e\; \backslash ,$

Derived:

1. $y=2x^3+6x^2-4x+\; \backslash \; frac\; \{9\}\; \{\backslash \; pi\}\; \backslash ,$

$y\text{'}=\; (2x^3)\; “+\; (6x^2)”\; -\; (4x)\; “+\; \backslash \; left\; (\backslash \; frac\; \{9\}\; \{\backslash \; pi\}\; \backslash \; right)”\; \backslash ,$

$y\text{'}=6x^2+12x-4+0\; \backslash ,$

$y\text{'}=2\; (3x^2+6x-2)\; \backslash ,$

2. $y=-x^3-5x^2+\; \backslash \; frac\; \{2\}\; \{3\}\; x-1\; \backslash ,$

$y\text{'}=-\; (x^3)\; “-\; (5x^2)\; “+\; \backslash \; left\; (\backslash \; frac\; \{2\}\; \{3\}\; X\; \backslash \; right)”\; -\; (1)”\; \backslash ,$

$y\text{'}=-3x^2-10x+\; \backslash \; frac\; \{2\}\; \{3\}\; -\; 0\; \backslash ,$

$y\text{'}=-3x^2-10x+\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash ,$

3. $y=\; \backslash \; frac\; \{5\}\; \{17\}\; x^3+x^2-2x+e\; \backslash ,$

$y\text{'}=\; \backslash \; left\; (\backslash \; frac\; \{5\}\; \{17\}\; x^3\; \backslash \; right)\; “+\; (x^2)”\; -\; (2x)\; “+\; (E)”\; \backslash ,$

$y\text{'}=\; \backslash \; frac\; \{5\; \backslash \; times\; 3x^2\}\; \{17\}\; +2x-2+0\; \backslash ,$

$y\text{'}=\; \backslash \; frac\; \{15x^2\}\; \{17\}\; +2x-2\; \backslash ,$

Unspecified degree

Are the general forms: $1)\; \backslash \; qquad\; y=f\; (X)\; =ax^b\; \backslash \; qquad\; (has\; \backslash \; not=0,\; B\; \backslash \; in\; \backslash \; mathbb\; \{N^*\})$

Nth derivative:

• $\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; \backslash \; qquad\; n=b:\; \backslash \; qquad\; y^\; \{(N)\}=a\; B!\; \backslash ,$

• $\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; \backslash \; qquad\; n>b:\; \backslash \; qquad\; y^\; \{(N)\}=0\; \backslash ,$

$2)\; \backslash \; qquad\; y=f\; (X)\; =ax^\; \{-\; B\}\; \backslash \; qquad\; (has\; \backslash \; not=0,\; B\; \backslash \; in\; \backslash \; mathbb\; \{N^*\})$

Nth derivative:

• $\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}:\; \backslash \; qquad\; y^\; \{(N)\}=a\; (-\; 1)\; ^n\; \backslash \; frac\; \{(b+n-1)!\}\{(b-1)!\; x^\{n+b\}\}$

Random links:

Perwez | Umberto Brunelleschi | Auzat | Beryl Markham | Bitty Schram | Dahus | Dacula,_la_Géorgie

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