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Step-by-step Solution

Find the derivative $\frac{d}{dx}\left(xe^{\left(3x-5\right)}+\ln\left(x-4\right)\right)$ using the sum rule

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Answer

$\frac{1}{x-4}+e^{\left(3x-5\right)}\left(1+3x\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(x e^{\left(3x-5\right)}+\ln\left(x-4\right)\right)$
1

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

$\frac{d}{dx}\left(xe^{\left(3x-5\right)}\right)+\frac{d}{dx}\left(\ln\left(x-4\right)\right)$
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Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=e^{\left(3x-5\right)}$

$e^{\left(3x-5\right)}\cdot\frac{d}{dx}\left(x\right)+x\frac{d}{dx}\left(e^{\left(3x-5\right)}\right)+\frac{d}{dx}\left(\ln\left(x-4\right)\right)$

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Answer

$\frac{1}{x-4}+e^{\left(3x-5\right)}\left(1+3x\right)$
$\frac{d}{dx}\left(x e^{\left(3x-5\right)}+\ln\left(x-4\right)\right)$

Main topic:

Equations

Used formulas:

7. See formulas

Time to solve it:

~ 0.74 seconds