👉 Probiere jetzt NerdPal aus! Unsere neue Mathe-App für iOS und Android
Taschenrechner AI-Whiteboard Arbeitsblätter & Tests
Lösungen buchen
Algebra Baldor
Premium gehen Themen
  1. Rechenmaschinen
  2. Ableitung Von Logarithmischen Funktionen

Ableitung von logarithmischen Funktionen Rechner

Mit unserem Ableitung von logarithmischen Funktionen 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.

Go!
Symbolischer Modus
Text-Modus
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Beispiel

Gelöste Probleme

Schwierige Probleme

1

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

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

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

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

Apply natural logarithm to both sides of the equality

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

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

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

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

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

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

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

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

$\frac{1}{y}\frac{d}{dx}\left(y\right)=\frac{d}{dx}\left(x+2\right)\ln\left(x\right)+\left(x+2\right)\frac{1}{x}\frac{d}{dx}\left(x\right)$
7

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

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

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

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

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

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

Multiply the fraction by the term

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

Any expression multiplied by $1$ is equal to itself

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

Multiply the fraction by the term

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

The derivative of the constant function ($2$) is equal to zero

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

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

$\frac{y^{\prime}}{y}=\ln\left(x\right)+\frac{x+2}{x}$
10

The derivative of a sum of two or more functions is the sum of the derivatives of each function

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

Multiply both sides of the equation by $y$

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

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

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

The derivative of the function results in

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

Endgültige Antwort auf das Problem

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

Haben Sie Probleme mit Mathematik?

Detaillierte Schritt-für-Schritt-Lösungen für Tausende von Problemen, die jeden Tag wachsen!

Beliebte Probleme

Die beliebtesten Probleme, die mit diesem Rechner gelöst wurden:

$\frac{d}{dx}\left(x^{x+2}\right)$ 72 Ansichten
$\frac{d}{dx}\left(\log_a\left(x\right)\right)$ 52 Ansichten
$\frac{d}{dx}\left(x^{x+7}\right)$ 52 Ansichten
$\frac{d}{dx}\left(x^{2\sqrt{x}}\right)$ 50 Ansichten
$\frac{d}{dx}\left(x-2\right)^{x+1}$ 48 Ansichten
$\frac{d}{dx}\left(\log_2\left(x\right)\right)$ 47 Ansichten
$\frac{d}{dx}\left(x^{x+9}\right)$ 44 Ansichten
$\frac{d}{dx}\left(4x\right)^{\ln\left(4x\right)}$ 43 Ansichten