Solve x^-frac{11{15}}div(x^-frac{7{15}}*x^-frac{13{15}})

$$x ^ { - \frac { 11 } { 15 } } \div ( x ^ { - \frac { 7 } { 15 } } \cdot x ^ { - \frac { 13 } { 15 } } )$$

Evaluate

$$x^{\frac{3}{5}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -\frac{7}{15} and -\frac{13}{15} to get -\frac{4}{3}.

$$\frac{x^{-\frac{11}{15}}}{x^{-\frac{4}{3}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{4}{3} from -\frac{11}{15} to get \frac{3}{5}.

$$x^{\frac{3}{5}}$$

Differentiate w.r.t. x

$$\frac{3}{5x^{\frac{2}{5}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -\frac{7}{15} and -\frac{13}{15} to get -\frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-\frac{11}{15}}}{x^{-\frac{4}{3}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract -\frac{4}{3} from -\frac{11}{15} to get \frac{3}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{5}})$$

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

$$\frac{3}{5}x^{\frac{3}{5}-1}$$

Subtract 1 from \frac{3}{5}.

$$\frac{3}{5}x^{-\frac{2}{5}}$$