Problem of the Week

Updated at Jan 16, 2017 1:47 PM

How can we solve for the derivative of 9xcos⁡x9x\cos{x}?

Below is the solution.



ddx9xcos⁡x\frac{d}{dx} 9x\cos{x}

1
Use Constant Factor Rule: ddxcf(x)=c(ddxf(x))\frac{d}{dx} cf(x)=c(\frac{d}{dx} f(x)).
9(ddxxcos⁡x)9(\frac{d}{dx} x\cos{x})

2
Use Product Rule to find the derivative of xcos⁡xx\cos{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
9((ddxx)cos⁡x+x(ddxcos⁡x))9((\frac{d}{dx} x)\cos{x}+x(\frac{d}{dx} \cos{x}))

3
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
9(cos⁡x+x(ddxcos⁡x))9(\cos{x}+x(\frac{d}{dx} \cos{x}))

4
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
9(cos⁡x−xsin⁡x)9(\cos{x}-x\sin{x})

Done
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