Solve (4t^{2}s^-3)^-1(8t^2s^4)

$$( 4 t ^ { 2 } s ^ { - 3 } ) ^ { - 1 } ( 8 t ^ { 2 } s ^ { 4 } )$$

Evaluate

$$2s^{7}$$

Solution Steps

Expand \left(4t^{2}s^{-3}\right)^{-1}.

$$4^{-1}\left(t^{2}\right)^{-1}\left(s^{-3}\right)^{-1}\times 8t^{2}s^{4}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$4^{-1}t^{-2}\left(s^{-3}\right)^{-1}\times 8t^{2}s^{4}$$

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

$$4^{-1}t^{-2}s^{3}\times 8t^{2}s^{4}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{1}{4}t^{-2}s^{3}\times 8t^{2}s^{4}$$

Multiply \frac{1}{4} and 8 to get 2.

$$2t^{-2}s^{3}t^{2}s^{4}$$

Multiply t^{-2} and t^{2} to get 1.

$$2s^{3}s^{4}$$

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

$$2s^{7}$$

Differentiate w.r.t. s

$$14s^{6}$$

Steps Using Definition of a Derivative

Expand \left(4t^{2}s^{-3}\right)^{-1}.

$$\frac{\mathrm{d}}{\mathrm{d}s}(4^{-1}\left(t^{2}\right)^{-1}\left(s^{-3}\right)^{-1}\times 8t^{2}s^{4})$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}s}(4^{-1}t^{-2}\left(s^{-3}\right)^{-1}\times 8t^{2}s^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}s}(4^{-1}t^{-2}s^{3}\times 8t^{2}s^{4})$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}s}(\frac{1}{4}t^{-2}s^{3}\times 8t^{2}s^{4})$$

Multiply \frac{1}{4} and 8 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}s}(2t^{-2}s^{3}t^{2}s^{4})$$

Multiply t^{-2} and t^{2} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}s}(2s^{3}s^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}s}(2s^{7})$$

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

$$7\times 2s^{7-1}$$

Multiply 7 times 2.

$$14s^{7-1}$$

Subtract 1 from 7.

$$14s^{6}$$