1-t^2/3+t^4

2. Re: Derivatives

The way that you've written this question is slightly ambiguous... You should use brackets.

Polynomial terms.
$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^{n} = nx^{n-1}$

Constants.
$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} k = 0$

Case#1
$\displaystyle \frac{\mathrm{d} }{\mathrm{d} t} -t^{\frac{2}{3}}+\frac{\mathrm{d} }{\mathrm{d} t}t^4 + \frac{\mathrm{d} }{\mathrm{d} t}1 = -\frac{2}{3t^{\frac{1}{3}}} + 4t^{3}$

Case#2
$\displaystyle \frac{\mathrm{d} }{\mathrm{d} t} \frac{-t^{2}}{3}+\frac{\mathrm{d} }{\mathrm{d} t}t^4 + \frac{\mathrm{d} }{\mathrm{d} t}1 = -\frac{2t}{3} + 4t^{3}$

Again, please use brackets to avoid ambiguity. :/

3. Re: Derivatives

Or even \displaystyle \displaystyle \begin{align*} \frac{d}{dt} \left( \frac{ 1 - t^2 }{ 3 + t^4 } \right) &= \frac{ -2t \left( 3 + t^4 \right) - 4t^3 \left( 1 - t^2 \right) }{ \left( 3 + t^4 \right) ^2} \end{align*}

Grammar is not just for English, and it's the difference between knowing your sh!t and knowing you're sh!t...