Find ds/dr r=1/(3s^2)-5/(2s)

Since ${\displaystyle \frac{1}{3}}$ is constant with respect to ${\displaystyle r}$, the derivative of ${\displaystyle \frac{1}{3s^2}}$ with respect to ${\displaystyle r}$ is ${\displaystyle \frac{1}{3}\frac{d}{dr}\left[\frac{1}{s^2}\right]}$.

Differentiate using the Quotient Rule which states that ${\displaystyle \frac{d}{dr}\left[\frac{f\left(r\right)}{g\left(r\right)}\right]}$ is ${\displaystyle \frac{g\left(r\right)\frac{d}{dr}\left[f\left(r\right)\right]-f\left(r\right)\frac{d}{dr}\left[g\left(r\right)\right]}{{g\left(r\right)}^2}}$ where ${\displaystyle f\left(r\right)=1}$ and ${\displaystyle g\left(r\right)=s^2}$.

Since ${\displaystyle 1}$ is constant with respect to ${\displaystyle r}$, the derivative of ${\displaystyle 1}$ with respect to ${\displaystyle r}$ is ${\displaystyle 0}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dr}\left[f\left(g\left(r\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(r\right)\right)g\prime \left(r\right)}$ where ${\displaystyle f\left(r\right)=r^2}$ and ${\displaystyle g\left(r\right)=s}$.

Rewrite ${\displaystyle \frac{d}{dr}\left[s\right]}$ as ${\displaystyle s\mathrm{<mo>\prime </mo>}}$.

Multiply ${\displaystyle s^2}$ by ${\displaystyle 0}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 1}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle -1}$.

Subtract ${\displaystyle 2ss\mathrm{<mo>\prime </mo>}}$ from ${\displaystyle 0}$.

Multiply the exponents in ${\displaystyle {\left(s^2\right)}^2}$.

Cancel the common factor of ${\displaystyle s}$ and ${\displaystyle s^4}$.

Move the negative in front of the fraction.

Multiply ${\displaystyle \frac{1}{3}}$ by ${\displaystyle \frac{2s\mathrm{<mo>\prime </mo>}}{s^3}}$.

Since ${\displaystyle -\frac{5}{2}}$ is constant with respect to ${\displaystyle r}$, the derivative of ${\displaystyle -\frac{5}{2s}}$ with respect to ${\displaystyle r}$ is ${\displaystyle -\frac{5}{2}\frac{d}{dr}\left[\frac{1}{s}\right]}$.

Differentiate using the Quotient Rule which states that ${\displaystyle \frac{d}{dr}\left[\frac{f\left(r\right)}{g\left(r\right)}\right]}$ is ${\displaystyle \frac{g\left(r\right)\frac{d}{dr}\left[f\left(r\right)\right]-f\left(r\right)\frac{d}{dr}\left[g\left(r\right)\right]}{{g\left(r\right)}^2}}$ where ${\displaystyle f\left(r\right)=1}$ and ${\displaystyle g\left(r\right)=s}$.

Since ${\displaystyle 1}$ is constant with respect to ${\displaystyle r}$, the derivative of ${\displaystyle 1}$ with respect to ${\displaystyle r}$ is ${\displaystyle 0}$.

Rewrite ${\displaystyle \frac{d}{dr}\left[s\right]}$ as ${\displaystyle s\mathrm{<mo>\prime </mo>}}$.

Multiply ${\displaystyle s}$ by ${\displaystyle 0}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 1}$.

Subtract ${\displaystyle s\mathrm{<mo>\prime </mo>}}$ from ${\displaystyle 0}$.

Move the negative in front of the fraction.

Multiply ${\displaystyle -1}$ by ${\displaystyle -1}$.

Multiply ${\displaystyle \frac{5}{2}}$ by ${\displaystyle 1}$.

Multiply ${\displaystyle \frac{5}{2}}$ by ${\displaystyle \frac{s\mathrm{<mo>\prime </mo>}}{s^2}}$.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle 2s^2,3s^3,1}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 2,3,1}$ then find LCM for the variable part ${\displaystyle s^2,s^3}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

Since ${\displaystyle 2}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 2}$.

Since ${\displaystyle 3}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 3}$.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

The LCM of ${\displaystyle 2,3,1}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

Multiply ${\displaystyle 2}$ by ${\displaystyle 3}$.

The factors for ${\displaystyle s^2}$ are ${\displaystyle s\cdot s}$, which is ${\displaystyle s}$ multiplied by each other ${\displaystyle 2}$ times.

The factors for ${\displaystyle s^3}$ are ${\displaystyle s\cdot s\cdot s}$, which is ${\displaystyle s}$ multiplied by each other ${\displaystyle 3}$ times.

The LCM of ${\displaystyle s^2,s^3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Simplify ${\displaystyle s\cdot s\cdot s}$.

The LCM for ${\displaystyle 2s^2,3s^3,1}$ is the numeric part ${\displaystyle 6}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{5s\mathrm{<mo>\prime </mo>}}{2s^2}-\frac{2s\mathrm{<mo>\prime </mo>}}{3s^3}=1}$ by ${\displaystyle 6s^3}$.

Simplify the left side.

Simplify the right side.

Factor ${\displaystyle s\mathrm{<mo>\prime </mo>}}$ out of ${\displaystyle 15s\mathrm{<mo>\prime </mo>}s-4s\mathrm{<mo>\prime </mo>}}$.

Divide each term in ${\displaystyle s\mathrm{<mo>\prime </mo>}(15s-4)=6s^3}$ by ${\displaystyle 15s-4}$ and simplify.