Solve (16x^-4)^-1/2div(8x^6)^-1/3

$$( 16 x ^ { - 4 } ) ^ { - 1 / 2 } \div ( 8 x ^ { 6 } ) ^ { - 1 / 3 }$$

Evaluate

$$\frac{x^{4}}{2}$$

Solution Steps

Expand \left(16x^{-4}\right)^{-\frac{1}{2}}.

$$\frac{16^{-\frac{1}{2}}\left(x^{-4}\right)^{-\frac{1}{2}}}{\left(8x^{6}\right)^{-\frac{1}{3}}}$$

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

$$\frac{16^{-\frac{1}{2}}x^{2}}{\left(8x^{6}\right)^{-\frac{1}{3}}}$$

Calculate 16 to the power of -\frac{1}{2} and get \frac{1}{4}.

$$\frac{\frac{1}{4}x^{2}}{\left(8x^{6}\right)^{-\frac{1}{3}}}$$

Expand \left(8x^{6}\right)^{-\frac{1}{3}}.

$$\frac{\frac{1}{4}x^{2}}{8^{-\frac{1}{3}}\left(x^{6}\right)^{-\frac{1}{3}}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -\frac{1}{3} to get -2.

$$\frac{\frac{1}{4}x^{2}}{8^{-\frac{1}{3}}x^{-2}}$$

Calculate 8 to the power of -\frac{1}{3} and get \frac{1}{2}.

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

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

$$\frac{\frac{1}{4}x^{4}}{\frac{1}{2}}$$

Divide \frac{1}{4}x^{4} by \frac{1}{2} by multiplying \frac{1}{4}x^{4} by the reciprocal of \frac{1}{2}.

$$\frac{1}{4}x^{4}\times 2$$

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

$$\frac{1}{2}x^{4}$$

Differentiate w.r.t. x

$$2x^{3}$$

Steps Using Definition of a Derivative

Expand \left(16x^{-4}\right)^{-\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16^{-\frac{1}{2}}\left(x^{-4}\right)^{-\frac{1}{2}}}{\left(8x^{6}\right)^{-\frac{1}{3}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16^{-\frac{1}{2}}x^{2}}{\left(8x^{6}\right)^{-\frac{1}{3}}})$$

Calculate 16 to the power of -\frac{1}{2} and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{4}x^{2}}{\left(8x^{6}\right)^{-\frac{1}{3}}})$$

Expand \left(8x^{6}\right)^{-\frac{1}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{4}x^{2}}{8^{-\frac{1}{3}}\left(x^{6}\right)^{-\frac{1}{3}}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and -\frac{1}{3} to get -2.

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

Calculate 8 to the power of -\frac{1}{3} and get \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{4}x^{2}}{\frac{1}{2}x^{-2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{4}x^{4}}{\frac{1}{2}})$$

Divide \frac{1}{4}x^{4} by \frac{1}{2} by multiplying \frac{1}{4}x^{4} by the reciprocal of \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{4}\times 2)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}x^{4})$$

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

$$4\times \frac{1}{2}x^{4-1}$$

Multiply 4 times \frac{1}{2}.

$$2x^{4-1}$$

Subtract 1 from 4.

$$2x^{3}$$