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Find the derivative $\frac{d}{dx}\left(\frac{e^x\cos\left(x\right)}{\tan\left(x\right)+\sin\left(x\right)}-e^x\ln\left(x\right)\right)$ using the sum rule

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Answer

$\frac{\left(e^x\cos\left(x\right)-e^x\sin\left(x\right)\right)\left(\tan\left(x\right)+\sin\left(x\right)\right)-e^x\cos\left(x\right)\left(\sec\left(x\right)^2+\cos\left(x\right)\right)}{\left(\tan\left(x\right)+\sin\left(x\right)\right)^2}-\left(e^x\ln\left(x\right)+e^x\frac{1}{x}\right)$

Step-by-step explanation

Problem to solve:

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

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

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

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

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

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Answer

$\frac{\left(e^x\cos\left(x\right)-e^x\sin\left(x\right)\right)\left(\tan\left(x\right)+\sin\left(x\right)\right)-e^x\cos\left(x\right)\left(\sec\left(x\right)^2+\cos\left(x\right)\right)}{\left(\tan\left(x\right)+\sin\left(x\right)\right)^2}-\left(e^x\ln\left(x\right)+e^x\frac{1}{x}\right)$
$\frac{d}{dx}\left(\frac{e^x\cos\left(x\right)}{\tan\left(x\right)+\sin\left(x\right)}-e^x\cdot\ln\left(x\right)\right)$

Main topic:

Sum rule

Used formulas:

6. See formulas

Time to solve it:

~ 0.65 seconds

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