Solve 3a^-1b^-7/3a^-7b^-1

$$\frac { 3 a ^ { - 1 } b ^ { - 7 } } { 3 a ^ { - 7 } b ^ { - 1 } }$$

Evaluate

$$\left(\frac{a}{b}\right)^{6}$$

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\frac{b^{-7}\times \frac{1}{a}}{a^{-7}\times \frac{1}{b}}$$

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

$$\frac{b^{-7}a^{6}}{\frac{1}{b}}$$

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

$$\frac{a^{6}}{b^{6}}$$

Differentiate w.r.t. a

$$\frac{6a^{5}}{b^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{b^{7}\times \frac{3}{b}}a^{-1-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{b^{6}}a^{6})$$

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}.

$$6\times \frac{1}{b^{6}}a^{6-1}$$

Do the arithmetic.

$$\frac{6}{b^{6}}a^{5}$$