Derive the function e^(-1x)ln(x) with respect to x

\frac{d}{dx}\left(e^{\left(-1\right)\cdot x}\ln\left(x\right)\right)

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Answer

$\frac{1}{x}e^{-x}-\ln\left(x\right)e^{-x}$

Step by step solution

Problem

$\frac{d}{dx}\left(e^{\left(-1\right)\cdot x}\ln\left(x\right)\right)$
1

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

$\frac{d}{dx}\left(\ln\left(x\right)\right)e^{-x}+\ln\left(x\right)\frac{d}{dx}\left(e^{-x}\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{x}e^{-x}\cdot\frac{d}{dx}\left(x\right)+\ln\left(x\right)\frac{d}{dx}\left(e^{-x}\right)$
3

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

$1\left(\frac{1}{x}\right)e^{-x}+\ln\left(x\right)\frac{d}{dx}\left(e^{-x}\right)$
4

Applying the derivative of the exponential function

$1\left(\frac{1}{x}\right)e^{-x}+1\ln\left(x\right)\frac{d}{dx}\left(-x\right)e^{-x}$
5

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

$1\left(\frac{1}{x}\right)e^{-x}+1\left(-1\right)\ln\left(x\right)e^{-x}\cdot\frac{d}{dx}\left(x\right)$
6

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

$1\left(\frac{1}{x}\right)e^{-x}+1\cdot 1\left(-1\right)\ln\left(x\right)e^{-x}$
7

Multiply $1$ times $-1$

$1\left(\frac{1}{x}\right)e^{-x}-\ln\left(x\right)e^{-x}$
8

Any expression multiplied by $1$ is equal to itself

$\frac{1}{x}e^{-x}-\ln\left(x\right)e^{-x}$

Answer

$\frac{1}{x}e^{-x}-\ln\left(x\right)e^{-x}$

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.21 seconds

Views:

110