Solve (X^1/2+1-2X^-1/2)(X^1/2-1-2X^-1/2)

$$( X ^ { 1 / 2 } + 1 - 2 X ^ { - 1 / 2 } ) ( X ^ { 1 / 2 } - 1 - 2 X ^ { - 1 / 2 } )$$

Evaluate

$$X-5+\frac{4}{X}$$

Solution Steps

Use the distributive property to multiply X^{\frac{1}{2}}+1-2X^{-\frac{1}{2}} by X^{\frac{1}{2}}-1-2X^{-\frac{1}{2}} and combine like terms.

$$\left(X^{\frac{1}{2}}\right)^{2}-4X^{\frac{1}{2}}X^{-\frac{1}{2}}-1+4X^{-1}$$

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

$$X^{1}-4X^{\frac{1}{2}}X^{-\frac{1}{2}}-1+4X^{-1}$$

Multiply X^{\frac{1}{2}} and X^{-\frac{1}{2}} to get 1.

$$X^{1}-4-1+4X^{-1}$$

Calculate X to the power of 1 and get X.

$$X-4-1+4X^{-1}$$

Subtract 1 from -4 to get -5.

$$X-5+4X^{-1}$$

Differentiate w.r.t. X

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

Steps Using Derivative Rule for Sum

Use the distributive property to multiply X^{\frac{1}{2}}+1-2X^{-\frac{1}{2}} by X^{\frac{1}{2}}-1-2X^{-\frac{1}{2}} and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}X}(\left(X^{\frac{1}{2}}\right)^{2}-4X^{\frac{1}{2}}X^{-\frac{1}{2}}-1+4X^{-1})$$

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

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

Multiply X^{\frac{1}{2}} and X^{-\frac{1}{2}} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}X}(X^{1}-4-1+4X^{-1})$$

Calculate X to the power of 1 and get X.

$$\frac{\mathrm{d}}{\mathrm{d}X}(X-4-1+4X^{-1})$$

Subtract 1 from -4 to get -5.

$$\frac{\mathrm{d}}{\mathrm{d}X}(X-5+4X^{-1})$$

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}.

$$X^{1-1}-4X^{-1-1}$$

Subtract 1 from 1.

$$X^{0}-4X^{-1-1}$$

Subtract 1 from -1.

$$X^{0}-4X^{-2}$$

For any term t except 0, t^{0}=1.

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