Solve (1-x^1/2)(1+x^frac{1{2}})

$$( 1 - x ^ { 1 / 2 } ) ( 1 + x ^ { \frac { 1 } { 2 } } )$$

Evaluate

$$1-x$$

Solution Steps

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

$$1-x^{1}$$

Calculate x to the power of 1 and get x.

$$1-x$$

Differentiate w.r.t. x

$$-1$$

Steps Using Derivative Rule for Product

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

$$\left(-\sqrt{x}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(\sqrt{x}+1)+\left(\sqrt{x}+1\right)\frac{\mathrm{d}}{\mathrm{d}x}(-\sqrt{x}+1)$$

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}.

$$\left(-\sqrt{x}+1\right)\times \frac{1}{2}x^{\frac{1}{2}-1}+\left(\sqrt{x}+1\right)\times \frac{1}{2}\left(-1\right)x^{\frac{1}{2}-1}$$

Simplify.

$$\left(-\sqrt{x}+1\right)\times \frac{1}{2}x^{-\frac{1}{2}}+\left(\sqrt{x}+1\right)\left(-\frac{1}{2}\right)x^{-\frac{1}{2}}$$

Multiply -\sqrt{x}+1 times \frac{1}{2}x^{-\frac{1}{2}}.

$$-\sqrt{x}\times \frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}+\left(\sqrt{x}+1\right)\left(-\frac{1}{2}\right)x^{-\frac{1}{2}}$$

Multiply \sqrt{x}+1 times -\frac{1}{2}x^{-\frac{1}{2}}.

$$-\sqrt{x}\times \frac{1}{2}x^{-\frac{1}{2}}+\frac{1}{2}x^{-\frac{1}{2}}+\sqrt{x}\left(-\frac{1}{2}\right)x^{-\frac{1}{2}}-\frac{1}{2}x^{-\frac{1}{2}}$$

To multiply powers of the same base, add their exponents.

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

Simplify.

$$-\frac{1}{2}x^{0}+\frac{1}{2}x^{-\frac{1}{2}}-\frac{1}{2}x^{0}-\frac{1}{2}x^{-\frac{1}{2}}$$

Combine like terms.

$$\frac{-1-1}{2}x^{0}+\frac{1-1}{2}x^{-\frac{1}{2}}$$

Add -\frac{1}{2} to -\frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-x^{0}+\frac{1-1}{2}x^{-\frac{1}{2}}$$

Add \frac{1}{2} to -\frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-x^{0}$$

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

$$-1$$