Solve 3b^-1/a^-2b^4*ab^3

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

Evaluate

$$\frac{3a}{b^{8}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

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

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

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

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

$$\frac{3}{\frac{1}{a}b^{8}}$$

Express \frac{1}{a}b^{8} as a single fraction.

$$\frac{3}{\frac{b^{8}}{a}}$$

Divide 3 by \frac{b^{8}}{a} by multiplying 3 by the reciprocal of \frac{b^{8}}{a}.

$$\frac{3a}{b^{8}}$$

Differentiate w.r.t. b

$$-\frac{24a}{b^{9}}$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{3a}{b^{3}}b^{-5})$$

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}.

$$-5\times \frac{3a}{b^{3}}b^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{15a}{b^{3}}\right)b^{-6}$$