# Find the derivative of the vector function. r(t)=a+tb+t^2c

Find the derivative of the vector function. $$\displaystyle{r}{\left({t}\right)}={a}+{t}{b}+{t}^{{2}}{c}$$

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Charles Randolph
$$\displaystyle{r}{\left({t}\right)}={a}+{t}{b}+{t}^{{2}}{c}$$
a,b,c are any constant vectors, like i, j, k, but not necessarily pointing in the same directions or having the same magnitudes.
Take the derivative of whatever is multiplying each vector, as usual.
$$\displaystyle{r}'{\left({t}\right)}={\left({1}\right)}'{a}+{\left({t}\right)}'{b}+{\left({t}^{{2}}\right)}'{c}$$
$$\displaystyle={\left({0}\right)}{a}+{\left({1}\right)}{b}+{\left({2}{t}\right)}{c}$$
$$\displaystyle={b}+{2}{t}{c}$$
Result: $$\displaystyle{r}'{\left({t}\right)}={K}+{2}{t}{c}$$