Question

Find the first and second derivatives. s=5t^{3}-3t^{5}

Derivatives
ANSWERED
asked 2021-06-06
Find the first and second derivatives.
\(\displaystyle{s}={5}{t}^{{{3}}}-{3}{t}^{{{5}}}\)

Answers (1)

2021-06-07
Step 1
Given function is
\(\displaystyle{s}={5}{t}^{{{3}}}-{3}{t}^{{{5}}}\)
Step 2
We will use power rule of differentiation which states that derivative of \(\displaystyle{x}^{{{n}}}\ {i}{s}\ {n}{x}^{{{\left({n}-{1}\right)}}}\)
\(\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}={\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({5}{t}^{{{3}}}\right)}-{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({3}{t}^{{{5}}}\right)}\)
\(\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}={5}{\left({3}{t}^{{{3}-{1}}}\right)}-{3}{\left({5}{t}^{{{5}-{1}}}\right)}\)
\(\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}={15}{t}^{{{2}}}-{15}{t}^{{{4}}}\)
Step 3
Again we will use power rule of differentiation to find second derivative
\(\displaystyle{\frac{{{d}^{{{2}}}{s}}}{{{\left.{d}{t}\right.}^{{{2}}}}}}={\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({15}{t}^{{{2}}}\right)}-{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({15}{t}^{{{4}}}\right)}\)
\(\displaystyle{\frac{{{d}^{{{2}}}{s}}}{{{\left.{d}{t}\right.}^{{{2}}}}}}={15}{\left({2}{t}^{{{2}-{1}}}\right)}-{15}{\left({4}{t}^{{{4}-{1}}}\right)}\)
\(\displaystyle{\frac{{{d}^{{{2}}}{s}}}{{{\left.{d}{t}\right.}^{{{2}}}}}}={30}{t}-{60}{t}^{{{3}}}\)
Step 4
Ans:
\(\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}={15}{t}^{{{2}}}-{15}{t}^{{{4}}}\)
\(\displaystyle{\frac{{{d}^{{{2}}}{s}}}{{{\left.{d}{t}\right.}^{{{2}}}}}}={30}{t}-{60}{t}^{{{3}}}\)
0
 
Best answer

expert advice

Need a better answer?
...