Solve x^frac{3{4}}(x^2x^frac{-1{3}})^frac{3{2}}

$$x ^ { \frac { 3 } { 4 } } ( x ^ { 2 } x ^ { \frac { - 1 } { 3 } } ) ^ { \frac { 3 } { 2 } }$$

Differentiate w.r.t. x

$$\frac{13x^{\frac{9}{4}}}{4}$$

Steps Using Definition of a Derivative

Fraction \frac{-1}{3} can be rewritten as -\frac{1}{3} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{4}}\left(x^{2}x^{-\frac{1}{3}}\right)^{\frac{3}{2}})$$

To multiply powers of the same base, add their exponents. Add 2 and -\frac{1}{3} to get \frac{5}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{4}}\left(x^{\frac{5}{3}}\right)^{\frac{3}{2}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{3} and \frac{3}{2} to get \frac{5}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{4}}x^{\frac{5}{2}})$$

To multiply powers of the same base, add their exponents. Add \frac{3}{4} and \frac{5}{2} to get \frac{13}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{13}{4}})$$

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

$$\frac{13}{4}x^{\frac{13}{4}-1}$$

Subtract 1 from \frac{13}{4}.

$$\frac{13}{4}x^{\frac{9}{4}}$$

Evaluate

$$x^{\frac{13}{4}}$$