Solve (x^{2})^-3x^-23(x^4)^-1/2

$$( x ^ { 2 } ) ^ { - 3 } x ^ { - 23 } ( x ^ { 4 } ) ^ { - 1 / 2 }$$

Evaluate

$$\frac{1}{x^{31}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and -3 to get -6.

$$x^{-6}x^{-23}\left(x^{4}\right)^{-\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{2} to get -2.

$$x^{-6}x^{-23}x^{-2}$$

To multiply powers of the same base, add their exponents. Add -6 and -23 to get -29.

$$x^{-29}x^{-2}$$

To multiply powers of the same base, add their exponents. Add -29 and -2 to get -31.

$$x^{-31}$$

Differentiate w.r.t. x

$$-\frac{31}{x^{32}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 2 and -3 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-6}x^{-23}\left(x^{4}\right)^{-\frac{1}{2}})$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{2} to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-6}x^{-23}x^{-2})$$

To multiply powers of the same base, add their exponents. Add -6 and -23 to get -29.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-29}x^{-2})$$

To multiply powers of the same base, add their exponents. Add -29 and -2 to get -31.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{-31})$$

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

$$-31x^{-31-1}$$

Subtract 1 from -31.

$$-31x^{-32}$$