Problem of the Week

Updated at Nov 9, 2015 4:42 PM

How can we find the derivative of xln⁡(x3)x\ln{({x}^{3})}?

Below is the solution.



ddxxln⁡(x3)\frac{d}{dx} x\ln{({x}^{3})}

1
Use Product Rule to find the derivative of xln⁡(x3)x\ln{({x}^{3})}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx)ln⁡(x3)+x(ddxln⁡(x3))(\frac{d}{dx} x)\ln{({x}^{3})}+x(\frac{d}{dx} \ln{({x}^{3})})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
ln⁡(x3)+x(ddxln⁡(x3))\ln{({x}^{3})}+x(\frac{d}{dx} \ln{({x}^{3})})

3
Use Chain Rule on ddxln⁡(x3)\frac{d}{dx} \ln{({x}^{3})}. Let u=x3u={x}^{3}. The derivative of ln⁡u\ln{u} is 1u\frac{1}{u}.
ln⁡(x3)+x(ddxx3)x3\ln{({x}^{3})}+\frac{x(\frac{d}{dx} {x}^{3})}{{x}^{3}}

4
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
ln⁡(x3)+3\ln{({x}^{3})}+3

Done
Cymath foriOSAndroid