Solve k^frac{1{2}}*k^frac{2{3}}divk^frac{3{4}}

$$k ^ { \frac { 1 } { 2 } } \times k ^ { \frac { 2 } { 3 } } \div k ^ { \frac { 3 } { 4 } }$$

Evaluate

$$k^{\frac{5}{12}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{2}{3} to get \frac{7}{6}.

$$\frac{k^{\frac{7}{6}}}{k^{\frac{3}{4}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{3}{4} from \frac{7}{6} to get \frac{5}{12}.

$$k^{\frac{5}{12}}$$

Differentiate w.r.t. k

$$\frac{5}{12k^{\frac{7}{12}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{2}{3} to get \frac{7}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{k^{\frac{7}{6}}}{k^{\frac{3}{4}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{3}{4} from \frac{7}{6} to get \frac{5}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(k^{\frac{5}{12}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{5}{12}k^{\frac{5}{12}-1}$$

Subtract 1 from \frac{5}{12}.

$$\frac{5}{12}k^{-\frac{7}{12}}$$