Question

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

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

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:
$$\displaystyle{\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}}={15}{t}^{{{2}}}-{15}{t}^{{{4}}}$$
$$\displaystyle{\frac{{{d}^{{{2}}}{s}}}{{{\left.{d}{t}\right.}^{{{2}}}}}}={30}{t}-{60}{t}^{{{3}}}$$