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  2. Differenziazione Avanzata

Calcolatrice di Differenziazione Avanzata

Risolvete i vostri problemi di matematica con la nostra calcolatrice Differenziazione Avanzata passo-passo. Migliorate le vostre abilità matematiche con il nostro ampio elenco di problemi impegnativi. Trova tutte le nostre calcolatrici qui.

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Esempio

Problemi risolti

Problemi difficili

1

Here, we show you a step-by-step solved example of advanced differentiation. This solution was automatically generated by our smart calculator:

$\frac{d}{dx}\left(2x\right)^x$
2

To derive the function $\left(2x\right)^x$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=\left(2x\right)^x$
3

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(\left(2x\right)^x\right)$
4

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=x\ln\left(2x\right)$
5

Derive both sides of the equality with respect to $x$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\ln\left(2x\right)\right)$
6

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\ln\left(2x\right)$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(x\right)\ln\left(2x\right)+x\frac{d}{dx}\left(\ln\left(2x\right)\right)$
7

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\ln\left(2x\right)+x\frac{d}{dx}\left(\ln\left(2x\right)\right)$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\ln\left(2x\right)+x\frac{1}{2x}\frac{d}{dx}\left(2x\right)$
8

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\ln\left(2x\right)+x\frac{1}{2x}\frac{d}{dx}\left(2x\right)$
9

The derivative of the linear function is equal to $1$

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+x\frac{1}{2x}\frac{d}{dx}\left(2x\right)$

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$2x\frac{1}{2x}\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$2x\frac{1}{2x}$
10

The derivative of the linear function times a constant, is equal to the constant

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+2x\frac{1}{2x}\frac{d}{dx}\left(x\right)$
11

The derivative of the linear function is equal to $1$

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+2x\frac{1}{2x}$

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+\frac{2\cdot 1x}{2x}$

Any expression multiplied by $1$ is equal to itself

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+\frac{2x}{2x}$

Simplify the fraction $\frac{2x}{2x}$ by $2x$

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+1$
12

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=\ln\left(2x\right)+1$
13

Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$

$\frac{y^{\prime}}{y}=\ln\left(2\right)+\ln\left(x\right)+1$
14

Multiply both sides of the equation by $y$

$y^{\prime}=\left(\ln\left(2\right)+\ln\left(x\right)+1\right)y$
15

Substitute $y$ for the original function: $\left(2x\right)^x$

$y^{\prime}=\left(\ln\left(2\right)+\ln\left(x\right)+1\right)\left(2x\right)^x$
16

The derivative of the function results in

$\left(\ln\left(2\right)+\ln\left(x\right)+1\right)\left(2x\right)^x$

Risposta finale al problema

$\left(\ln\left(2\right)+\ln\left(x\right)+1\right)\left(2x\right)^x$

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