Four elementary and composite derivatives

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Compute the derivative of the following functions, indicating the composition scheme when the function is composite.

i) $f(x)=2x^5+1$; ii) $f(x)=x\sin x$; iii) $f(x)=e^{x^2}$; iv) $f(x)=\ln(2x)$.

Solution

i) Polynomial: $f'(x)=10x^4.$

ii) Product rule: $f'(x)=1\cdot\sin x+x\cos x=\sin x+x\cos x.$

iii) Composite $e^{u}$ with $u=x^2$, $u'=2x$: $f'(x)=e^{x^2}\cdot 2x=2x\,e^{x^2}.$

iv) Composite $\ln(u)$ with $u=2x$, $u'=2$ (or $\ln(2x)=\ln2+\ln x$): $f'(x)=\frac{1}{2x}\cdot 2=\frac1x.$

Summary of results: $\boxed{\;10x^4,\quad \sin x+x\cos x,\quad 2x\,e^{x^2},\quad \tfrac1x\;}$