Problem of the Week

Updated at May 8, 2017 2:54 PM

To get more practice in calculus, we brought you this problem of the week:

How can we find the derivative of 4xcsc⁡x4x\csc{x}?

Check out the solution below!



ddx4xcsc⁡x\frac{d}{dx} 4x\csc{x}

1
Use Constant Factor Rule: ddxcf(x)=c(ddxf(x))\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x)).
4(ddxxcsc⁡x)4(\frac{d}{dx} x\csc{x})

2
Use Product Rule to find the derivative of xcsc⁡xx\csc{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
4((ddxx)csc⁡x+x(ddxcsc⁡x))4((\frac{d}{dx} x)\csc{x}+x(\frac{d}{dx} \csc{x}))

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
4(csc⁡x+x(ddxcsc⁡x))4(\csc{x}+x(\frac{d}{dx} \csc{x}))

4
Use Trigonometric Differentiation: the derivative of csc⁡x\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
4(csc⁡x−xcsc⁡xcot⁡x)4(\csc{x}-x\csc{x}\cot{x})

Done
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