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\frac{d}{dx}\left(\frac{e^x\cos\left(x\right)}{\tan\left(x\right)+\sin\left(x\right)}-e^x\cdot\ln\left(x\right)\right)

Find the derivative of (e^xcos(x))/(tan(x)+sin(x))e^xln(x)*-1

Answer

$\frac{x\left(\left(\sin\left(x\right)+\tan\left(x\right)\right)\left(e^x\cos\left(x\right)-e^x\sin\left(x\right)\right)-e^x\left(\cos\left(x\right)+\sec\left(x\right)^2\right)\cos\left(x\right)\right)-\left(\sin\left(x\right)+\tan\left(x\right)\right)^2e^x}{\left(\sin\left(x\right)+\tan\left(x\right)\right)^2x}-e^x\ln\left(x\right)$

Step-by-step explanation

Problem

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

La derivada de la suma de dos o más funciones equivale a la suma de la derivada de cada función por separado

$\frac{d}{dx}\left(-e^x\ln\left(x\right)\right)+\frac{d}{dx}\left(\frac{e^x\cos\left(x\right)}{\sin\left(x\right)+\tan\left(x\right)}\right)$

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Answer

$\frac{x\left(\left(\sin\left(x\right)+\tan\left(x\right)\right)\left(e^x\cos\left(x\right)-e^x\sin\left(x\right)\right)-e^x\left(\cos\left(x\right)+\sec\left(x\right)^2\right)\cos\left(x\right)\right)-\left(\sin\left(x\right)+\tan\left(x\right)\right)^2e^x}{\left(\sin\left(x\right)+\tan\left(x\right)\right)^2x}-e^x\ln\left(x\right)$

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$\frac{d}{dx}\left(\frac{e^x\cos\left(x\right)}{\tan\left(x\right)+\sin\left(x\right)}-e^x\cdot\ln\left(x\right)\right)$

Main topic:

Product rule

Used formulas:

6. See formulas

Time to solve it:

~ 5.41 seconds