Three derivatives with the chain rule

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

i) $f(x)=x^6+5x^3$; ii) $f(x)=\sin(e^x)$; iii) $f(x)=\sqrt{\tan x}$.

Solution

i) Polynomial, term-by-term derivative: $f'(x)=6x^5+15x^2.$

ii) Composite $\sin(u)$ with $u=e^x$, $u'=e^x$: $f'(x)=\cos(u)\,u'=e^x\cos(e^x).$

iii) $\sqrt{\tan x}=(\tan x)^{1/2}$, composite with $u=\tan x$, $u'=\sec^2 x=\dfrac1{\cos^2 x}$: $f'(x)=\tfrac12(\tan x)^{-1/2}\sec^2 x=\frac{\sec^2 x}{2\sqrt{\tan x}}=\frac{1}{2\sqrt{\tan x}\,\cos^2 x}.$

The most involved result is the one in part iii): $\boxed{\;f'(x)=\dfrac{\sec^2 x}{2\sqrt{\tan x}}\;}$