rcih compute derivatives for sums, constant multiples, and power, polynomial, trig, exponen- tial, logarithmic, and inverse trigonometric functions using shortcuts. Ose the short cut rules to compute the following derivatives. Simplify algebraically first so thát all of your terms are painfully obvious power functions, and then apply the power rule. (a) f(x) = x - 5x+ 12 NG (b) f(t) = (c) f(x) = V4r3 + 6x7 – 2

(6) (a) Given function is $f\left(x\right)=x^2-5x^{-1}+12$

Use power rule formula $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ and $\frac{d}{dx}\left(c\right)=0$, where $c$ is constant.

$\begin{array}{c}{f}^{\text{'}}\left(x\right)=\frac{d}{dx}\left(x^2-5x^{-1}+12\right)\\ =\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-5x^{-1}\right)+\frac{d}{dx}\left(12\right)\\ =\frac{d}{dx}\left(x^2\right)-5\frac{d}{dx}\left(x^{-1}\right)+\frac{d}{dx}\left(12\right)\\ =2x^{2-1}-5\cdot \left(-1\right)x^{-1-1}+0\\ =2x+5x^{-2}\end{array}$

Hence, $f^{\text{'}}\left(x\right)=2x+5x^{-2}$
(b) Given function is $f\left(t\right)=\frac{2}{3t^{\frac{1}{2}}}$

It can be written as $f\left(t\right)=\frac{2}{3}{t}^{-\frac{1}{2}}$

$\begin{array}{c}{f}^{\text{'}}\left(t\right)=\frac{2}{3}\left(-\frac{1}{2}\right)t^{-\frac{1}{2}-1}\\ =-\frac{1}{3}{t}^{-\frac{3}{2}}\end{array}$

Hence, $f^{\text{'}}\left(t\right)=-\frac{1}{3}{t}^{-\frac{3}{2}}$