Solve (r^frac{8{7}}divr^frac{5{3}})divr^frac{5{6}}

$$( r ^ { \frac { 8 } { 7 } } \div r ^ { \frac { 5 } { 3 } } ) \div r ^ { \frac { 5 } { 6 } }$$

Differentiate w.r.t. r

$$-\frac{19}{14r^{\frac{33}{14}}}$$

Steps Using Chain Rule

Rewrite r^{\frac{5}{3}} as r^{\frac{8}{7}}r^{\frac{11}{21}}. Cancel out r^{\frac{8}{7}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{\frac{1}{r^{\frac{11}{21}}}}{r^{\frac{5}{6}}})$$

Express \frac{\frac{1}{r^{\frac{11}{21}}}}{r^{\frac{5}{6}}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{r^{\frac{11}{21}}r^{\frac{5}{6}}})$$

To multiply powers of the same base, add their exponents. Add \frac{11}{21} and \frac{5}{6} to get \frac{19}{14}.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{r^{\frac{19}{14}}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(r^{\frac{19}{14}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}r}(r^{\frac{19}{14}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(r^{\frac{19}{14}}\right)^{-2}\times \frac{19}{14}r^{\frac{19}{14}-1}$$

Simplify.

$$-\frac{19}{14}r^{\frac{5}{14}}\left(r^{\frac{19}{14}}\right)^{-2}$$

Evaluate

$$\frac{1}{r^{\frac{19}{14}}}$$