Solve frac{9a^{3}b^{-1/3}c^frac{-1{2}}}{3a^-frac{1{3}}b^-6c^4}

$$\frac { 9 a ^ { 3 } b ^ { - \frac { 1 } { 3 } } c ^ { \frac { - 1 } { 2 } } } { 3 a ^ { - \frac { 1 } { 3 } } b ^ { - 6 } c ^ { 4 } }$$

Differentiate w.r.t. a

$$\frac{10a^{\frac{7}{3}}b^{\frac{17}{3}}}{c^{\frac{9}{2}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{9}{\sqrt[3]{b}\sqrt{c}\times \frac{3c^{4}}{b^{6}}}a^{3-\left(-\frac{1}{3}\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3b^{\frac{17}{3}}}{c^{\frac{9}{2}}}a^{\frac{10}{3}})$$

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}.

$$\frac{10}{3}\times \frac{3b^{\frac{17}{3}}}{c^{\frac{9}{2}}}a^{\frac{10}{3}-1}$$

Do the arithmetic.

$$\frac{10b^{\frac{17}{3}}}{c^{\frac{9}{2}}}a^{\frac{7}{3}}$$

Evaluate

$$\frac{3a^{\frac{10}{3}}b^{\frac{17}{3}}}{c^{\frac{9}{2}}}$$