Question

Find the indicated derivatives. \frac{dr}{ds} if r=s^{3}-2s^{2}+3

Derivatives
ANSWERED
asked 2021-03-05
Find the indicated derivatives. \(\displaystyle{\frac{{{d}{r}}}{{{d}{s}}}}{\quad\text{if}\quad}{r}={s}^{{{3}}}-{2}{s}^{{{2}}}+{3}\)

Expert Answers (1)

2021-03-07
Step 1
Given: \(\displaystyle{r}={s}^{{{3}}}-{2}{s}^{{{2}}}+{3}\)
for finding value of \(\displaystyle{\frac{{{d}{r}}}{{{d}{s}}}}\), we differentiate given function with respect to s
Step 2
so,
\(\displaystyle{\frac{{{d}{r}}}{{{\left.{d}{x}\right.}}}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({s}^{{{3}}}-{2}{s}^{{{2}}}+{3}\right)}\)
\(\displaystyle={\frac{{{d}}}{{{d}{s}}}}{\left({s}^{{{3}}}\right)}-{2}{\frac{{{d}}}{{{d}{s}}}}{\left({s}^{{{2}}}\right)}+{\frac{{{d}}}{{{d}{s}}}}{\left({3}\right)}\)
\(\displaystyle{\left(\because{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{n}}}\right)}={n}{x}^{{{n}-{1}}}\right)}\)
\(\displaystyle={3}{s}^{{{2}}}-{2}{\left({2}{s}\right)}+{0}\)
\(\displaystyle={3}{s}^{{{2}}}-{4}{s}\)
hence, \(\displaystyle{\frac{{{d}{r}}}{{{d}{s}}}}\) is equal to \(\displaystyle{3}{s}^{{{2}}}-{4}{s}\).
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