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  1. Rechenmaschinen
  2. Erweiterte Differenzierung

Erweiterte Differenzierung Rechner

Mit unserem Erweiterte Differenzierung Schritt-für-Schritt-Rechner erhalten Sie detaillierte Lösungen für Ihre mathematischen Probleme. Üben Sie Ihre mathematischen Fähigkeiten und lernen Sie Schritt für Schritt mit unserem Mathe-Löser. Alle unsere Online-Rechner finden Sie hier.

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Beispiel

Gelöste Probleme

Schwierige Probleme

1

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

$\frac{d}{dx}\left(x^{7\sin\left(x\right)}\right)$
2

To derive the function $x^{7\sin\left(x\right)}$, use the method of logarithmic differentiation. First, assign the function to $y$, then take the natural logarithm of both sides of the equation

$y=x^{7\sin\left(x\right)}$
3

Apply natural logarithm to both sides of the equality

$\ln\left(y\right)=\ln\left(x^{7\sin\left(x\right)}\right)$
4

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

$\ln\left(y\right)=7\sin\left(x\right)\ln\left(x\right)$
5

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=\frac{d}{dx}\left(7\sin\left(x\right)\ln\left(x\right)\right)$
6

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

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

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

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\left(\frac{d}{dx}\left(\sin\left(x\right)\right)\ln\left(x\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\sin\left(x\right)\right)$
8

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{d}{dx}\left(\ln\left(y\right)\right)=7\left(\frac{d}{dx}\left(x\right)\cos\left(x\right)\ln\left(x\right)+\frac{d}{dx}\left(\ln\left(x\right)\right)\sin\left(x\right)\right)$
9

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

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

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{1}{x}\frac{d}{dx}\left(x\right)\sin\left(x\right)\right)$
10

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{y}\frac{d}{dx}\left(y\right)=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{1}{x}\frac{d}{dx}\left(x\right)\sin\left(x\right)\right)$

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

$\frac{y^{\prime}}{y}=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{1}{x}\sin\left(x\right)\right)$
11

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

$\frac{y^{\prime}}{y}=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{1}{x}\sin\left(x\right)\right)$

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{1\sin\left(x\right)}{x}\right)$

Any expression multiplied by $1$ is equal to itself

$\frac{y^{\prime}}{y}=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{\sin\left(x\right)}{x}\right)$
12

Multiply the fraction by the term

$\frac{y^{\prime}}{y}=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{\sin\left(x\right)}{x}\right)$
13

Multiply both sides of the equation by $y$

$y^{\prime}=7y\left(\cos\left(x\right)\ln\left(x\right)+\frac{\sin\left(x\right)}{x}\right)$
14

Substitute $y$ for the original function: $x^{7\sin\left(x\right)}$

$y^{\prime}=7\left(\cos\left(x\right)\ln\left(x\right)+\frac{\sin\left(x\right)}{x}\right)x^{7\sin\left(x\right)}$
15

The derivative of the function results in

$7\left(\cos\left(x\right)\ln\left(x\right)+\frac{\sin\left(x\right)}{x}\right)x^{7\sin\left(x\right)}$

Endgültige Antwort auf das Problem

$7\left(\cos\left(x\right)\ln\left(x\right)+\frac{\sin\left(x\right)}{x}\right)x^{7\sin\left(x\right)}$

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Verwandte Rechner

Grundlegende Differenzierungsregeln Rechner

Beliebte Probleme

Die beliebtesten Probleme, die mit diesem Rechner gelöst wurden:

$\frac{d}{dx}\left(x^{7\sin\left(x\right)}\right)$ 147 Ansichten
$\frac{d}{dx}\left(x^{x^3}\right)$ 123 Ansichten
$\frac{d}{dx}\left(x^{x+2}\right)$ 72 Ansichten
$\frac{d}{dx}\left(2x\right)^x$ 69 Ansichten
$\frac{d}{dx}\left(tan^{9-7x}\left(x\right)\right)$ 67 Ansichten
$\frac{d}{dx}\left(1+\frac{1}{x}\right)^x$ 67 Ansichten
$\frac{d}{dx}\left(x^{x+6}\right)$ 61 Ansichten
$\frac{d}{dx}\left(x^{x+7}\right)$ 52 Ansichten