Solve r^-frac{7{3}}divr^-frac{5{2}}

$$r ^ { - \frac { 7 } { 3 } } \div r ^ { - \frac { 5 } { 2 } }$$

Differentiate w.r.t. r

$$\frac{1}{6r^{\frac{5}{6}}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{1}r^{-\frac{7}{3}-\left(-\frac{5}{2}\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\sqrt[6]{r})$$

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}.

$$\frac{1}{6}r^{\frac{1}{6}-1}$$

Do the arithmetic.

$$\frac{1}{6}r^{-\frac{5}{6}}$$

Evaluate

$$\sqrt[6]{r}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{5}{2} from -\frac{7}{3} to get \frac{1}{6}.

$$r^{\frac{1}{6}}$$