Solve (sqrt{x}+sqrt{x-alpha^2})(sqrt{x}-sqrt{x-alpha^2})

$$( \sqrt { x } + \sqrt { x - \alpha ^ { 2 } } ) ( \sqrt { x } - \sqrt { x - \alpha ^ { 2 } } )$$

Evaluate

$$\alpha ^{2}$$

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}.

$$\left(\sqrt{x}\right)^{2}-\left(\sqrt{x-\alpha ^{2}}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$x-\left(\sqrt{x-\alpha ^{2}}\right)^{2}$$

Calculate \sqrt{x-\alpha ^{2}} to the power of 2 and get x-\alpha ^{2}.

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

To find the opposite of x-\alpha ^{2}, find the opposite of each term.

$$x-x+\alpha ^{2}$$

Combine x and -x to get 0.

$$\alpha ^{2}$$

Differentiate w.r.t. α

$$2\alpha$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\left(\sqrt{x}\right)^{2}-\left(\sqrt{x-\alpha ^{2}}\right)^{2})$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(x-\left(\sqrt{x-\alpha ^{2}}\right)^{2})$$

Calculate \sqrt{x-\alpha ^{2}} to the power of 2 and get x-\alpha ^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(x-\left(x-\alpha ^{2}\right))$$

To find the opposite of x-\alpha ^{2}, find the opposite of each term.

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

Combine x and -x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\alpha ^{2})$$

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

$$2\alpha ^{2-1}$$

Subtract 1 from 2.

$$2\alpha ^{1}$$

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

$$2\alpha$$