Answer
$ \displaystyle \frac{dy}{dx}=0.4x^{0.2}-0.45x^{-0.1}$
Work Step by Step
SUMMARY (rules in differential notation):
1. The Power Rule$:\ \ \ \displaystyle \frac{d}{dx}[x^{n}]=n\cdot x^{n-1 } $
2. Sum Rule: $\displaystyle \ \ \ \frac{d}{dx}[f\pm g](x)=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)] $
3. Constant Multiple Rule:$\ \ \displaystyle \frac{d}{dx}[cf(x)]=c\cdot\frac{d}{dx}[f(x)] $
4. Constant times x:$\ \ \ \displaystyle \frac{d}{dx}(cx)=c $
5. Constant:$\displaystyle \ \ \ \ \ \frac{d}{dx}(c)=0 $
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$ \displaystyle \frac{dy}{dx}= \frac{d}{dx}[ \frac{1}{3}x^{1.2}-\frac{1}{2}x^{0.9})$ = $\ \ \ $...(2)
$=\displaystyle \frac{d}{dx}( \frac{1}{3}x^{1.2})-\frac{d}{dx}(\frac{1}{2}x^{0.9})$ = $\ \ \ $...($3$)
$=\displaystyle \frac{1}{3}\frac{d}{dx}( x^{1.2})-\frac{1}{2}\frac{d}{dx}(x^{0.9})$ = $\ \ \ $...($1$)
$=\displaystyle \frac{1}{3}(1.2x^{0.2})-\frac{1}{2}(0.9x^{-0.1})$
$ \displaystyle \frac{dy}{dx}=0.4x^{0.2}-0.45x^{-0.1}$
