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Step-by-step Solution

Derive the function cot(3*x)+xsin(x^3+5) with respect to x

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Answer

$-3\csc\left(3x\right)^2+\sin\left(x^3+5\right)+3x^{3}\cos\left(x^3+5\right)$

Step-by-step explanation

Problem to solve:

$\frac{d}{dx}\left(\cot\left(3x\right)+x\cdot\sin\left(x^3+5\right)\right)$
1

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

$\frac{d}{dx}\left(\cot\left(3x\right)\right)+\frac{d}{dx}\left(x\sin\left(x^3+5\right)\right)$
2

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=\sin\left(x^3+5\right)$

$\frac{d}{dx}\left(\cot\left(3x\right)\right)+\frac{d}{dx}\left(x\right)\sin\left(x^3+5\right)+x\frac{d}{dx}\left(\sin\left(x^3+5\right)\right)$

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Answer

$-3\csc\left(3x\right)^2+\sin\left(x^3+5\right)+3x^{3}\cos\left(x^3+5\right)$

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$\frac{d}{dx}\left(\cot\left(3x\right)+x\cdot\sin\left(x^3+5\right)\right)$

Main topic:

Differential calculus

Used formulas:

6. See formulas

Time to solve it:

~ 0.47 seconds