Solve 4a^frac{2{3}}*b^-frac{3{3}}div(-2/3a^-frac{2{3}}b^frac{3{3}})

$$4 a ^ { \frac { 2 } { 3 } } \cdot b ^ { - \frac { 3 } { 3 } } \div ( - \frac { 2 } { 3 } a ^ { - \frac { 2 } { 3 } } b ^ { \frac { 3 } { 3 } } )$$

Evaluate

$$-\frac{6a^{\frac{4}{3}}}{b^{2}}$$

Solution Steps

Divide 3 by 3 to get 1.

$$\frac{4a^{\frac{2}{3}}b^{-1}}{-\frac{2}{3}a^{-\frac{2}{3}}b^{\frac{3}{3}}}$$

Divide 3 by 3 to get 1.

$$\frac{4a^{\frac{2}{3}}b^{-1}}{-\frac{2}{3}a^{-\frac{2}{3}}b^{1}}$$

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

$$\frac{4b^{-1}a^{\frac{4}{3}}}{-\frac{2}{3}b^{1}}$$

Calculate b to the power of 1 and get b.

$$\frac{4b^{-1}a^{\frac{4}{3}}}{-\frac{2}{3}b}$$

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

$$\frac{4a^{\frac{4}{3}}}{-\frac{2}{3}b^{2}}$$

Differentiate w.r.t. a

$$-\frac{8\sqrt[3]{a}}{b^{2}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(-\frac{6}{b^{2}}\right)a^{\frac{4}{3}})$$

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}.

$$\frac{4}{3}\left(-\frac{6}{b^{2}}\right)a^{\frac{4}{3}-1}$$

Do the arithmetic.

$$\left(-\frac{8}{b^{2}}\right)\sqrt[3]{a}$$