Question

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

Derivatives
Find the indicated derivatives. $$\displaystyle{\frac{{{d}{r}}}{{{d}{s}}}}{\quad\text{if}\quad}{r}={s}^{{{3}}}-{2}{s}^{{{2}}}+{3}$$

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 $$\displaystyle{3}{s}^{{{2}}}-{4}{s}$$.