# Logarithmic differentiation Calculator

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### Difficult Problems

1

Example

$\frac{d}{dx}\left(\ln\left(x\sqrt{a+x}\right)\right)$
2

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

$\frac{1}{\sqrt{x+a}x}\cdot\frac{d}{dx}\left(\sqrt{x+a}x\right)$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\sqrt{x+a}$

$\frac{1}{\sqrt{x+a}x}\left(x\frac{d}{dx}\left(\sqrt{x+a}\right)+\sqrt{x+a}\cdot\frac{d}{dx}\left(x\right)\right)$
4

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

$\frac{1}{\sqrt{x+a}x}\left(x\frac{d}{dx}\left(\sqrt{x+a}\right)+\sqrt{x+a}\right)$
5

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{1}{\sqrt{x+a}x}\left(\frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}\cdot\frac{d}{dx}\left(x+a\right)+\sqrt{x+a}\right)$
6

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

$\frac{1}{\sqrt{x+a}x}\left(\frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(a\right)\right)+\sqrt{x+a}\right)$
7

The derivative of the constant function is equal to zero

$\frac{1}{\sqrt{x+a}x}\left(\frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(x\right)+0\right)+\sqrt{x+a}\right)$
8

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

$\left(\left(1+0\right)\cdot \frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}+\sqrt{x+a}\right)\frac{1}{\sqrt{x+a}x}$
9

Add the values $0$ and $1$

$\left(1\cdot \frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}+\sqrt{x+a}\right)\frac{1}{\sqrt{x+a}x}$
10

Multiply $\frac{1}{2}$ times $1$

$\left(\frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}+\sqrt{x+a}\right)\frac{1}{\sqrt{x+a}x}$
11

Multiplying the fraction and term

$\frac{\frac{1}{2}x\left(x+a\right)^{-\frac{1}{2}}+\sqrt{x+a}}{\sqrt{x+a}x}$
12

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{\frac{1}{2}x\frac{1}{\sqrt{x+a}}+\sqrt{x+a}}{\sqrt{x+a}x}$
13

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}, where a=\frac{1}{2} and x=\sqrt{x+a} \frac{x\frac{\frac{1}{2}}{\sqrt{x+a}}+\sqrt{x+a}}{\sqrt{x+a}x} 14 Multiplying the fraction and term \frac{\frac{\frac{1}{2}x}{\sqrt{x+a}}+\sqrt{x+a}}{\sqrt{x+a}x} 15 Apply the formula: \frac{b}{c}+a$$=\frac{b+c\cdot a}{c}$, where $a=\sqrt{x+a}$, $b=\frac{1}{2}x$ and $c=\sqrt{x+a}$

$\frac{\frac{\frac{1}{2}x+\sqrt{x+a}\sqrt{x+a}}{\sqrt{x+a}}}{\sqrt{x+a}x}$
16

When multiplying exponents with same base we can add the exponents

$\frac{\frac{\frac{1}{2}x+x+a}{\sqrt{x+a}}}{\sqrt{x+a}x}$
17

Adding $\frac{1}{2}x$ and $x$

$\frac{\frac{a+\frac{3}{2}x}{\sqrt{x+a}}}{\sqrt{x+a}x}$
18

Simplifying the fraction

$\frac{a+\frac{3}{2}x}{\sqrt{x+a}\sqrt{x+a}x}$
19

When multiplying exponents with same base we can add the exponents

$\frac{a+\frac{3}{2}x}{x\left(x+a\right)}$
20

Multiplying polynomials $x$ and $a+x$

$\frac{a+\frac{3}{2}x}{x^2+a\cdot x}$