Solve (3a^2b^-1*2ab)^-1

$$( 3 a ^ { 2 } b ^ { - 1 } \cdot 2 a b ) ^ { - 1 }$$

Evaluate

$$\frac{1}{6a^{3}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\left(3a^{3}b^{-1}\times 2b\right)^{-1}$$

Multiply b^{-1} and b to get 1.

$$\left(3a^{3}\times 2\right)^{-1}$$

Multiply 3 and 2 to get 6.

$$\left(6a^{3}\right)^{-1}$$

Expand \left(6a^{3}\right)^{-1}.

$$6^{-1}\left(a^{3}\right)^{-1}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$6^{-1}a^{-3}$$

Calculate 6 to the power of -1 and get \frac{1}{6}.

$$\frac{1}{6}a^{-3}$$

Differentiate w.r.t. a

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

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Multiply b^{-1} and b to get 1.

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

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(6a^{3}\right)^{-1})$$

Expand \left(6a^{3}\right)^{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(6^{-1}\left(a^{3}\right)^{-1})$$

To raise a power to another power, multiply the exponents. Multiply 3 and -1 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}a}(6^{-1}a^{-3})$$

Calculate 6 to the power of -1 and get \frac{1}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{6}a^{-3})$$

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

$$-3\times \frac{1}{6}a^{-3-1}$$

Multiply -3 times \frac{1}{6}.

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

Subtract 1 from -3.

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