Question

# Find the indicated derivatives. \frac{ds}{dt} if s=\frac{t}{2t+1}

Derivatives
Find the indicated derivatives. $$\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}{\quad\text{if}\quad}{s}={\frac{{{t}}}{{{2}{t}+{1}}}}$$

2021-04-22
Step 1
The given function is $$\displaystyle{s}={\frac{{{t}}}{{{2}{t}+{1}}}}$$
Step 2
Find the derivative $$\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}$$
$$\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}={\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({\frac{{{t}}}{{{2}{t}+{1}}}}\right)}$$
$$\displaystyle={\frac{{{\left({2}{t}+{1}\right)}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({t}\right)}-{t}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({2}{t}+{1}\right)}}}{{{\left({2}{t}+{1}\right)}^{{{2}}}}}}$$
$$\displaystyle={\frac{{{\left({2}{t}+{1}\right)}{\left({1}\right)}-{t}{\left({2}\right)}}}{{{\left({2}{t}+{1}\right)}^{{{2}}}}}}$$
$$\displaystyle={\frac{{{2}{t}+{1}-{2}{t}}}{{{\left({2}{t}+{1}\right)}^{{{2}}}}}}$$
$$\displaystyle={\frac{{{1}}}{{{\left({2}{t}+{1}\right)}^{{{2}}}}}}$$