Solve (sqrt{m+n}+sqrt{m-n})(sqrt{m+n}-sqrt{m-n})

$$( \sqrt { m + n } + \sqrt { m - n } ) ( \sqrt { m + n } - \sqrt { m - n } )$$

Evaluate

$$2n$$

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}.

$$\left(\sqrt{m+n}\right)^{2}-\left(\sqrt{m-n}\right)^{2}$$

Calculate \sqrt{m+n} to the power of 2 and get m+n.

$$m+n-\left(\sqrt{m-n}\right)^{2}$$

Calculate \sqrt{m-n} to the power of 2 and get m-n.

$$m+n-\left(m-n\right)$$

To find the opposite of m-n, find the opposite of each term.

$$m+n-m-\left(-n\right)$$

The opposite of -n is n.

$$m+n-m+n$$

Combine m and -m to get 0.

$$n+n$$

Combine n and n to get 2n.

$$2n$$

Differentiate w.r.t. n

$$2$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\left(\sqrt{m+n}\right)^{2}-\left(\sqrt{m-n}\right)^{2})$$

Calculate \sqrt{m+n} to the power of 2 and get m+n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(m+n-\left(\sqrt{m-n}\right)^{2})$$

Calculate \sqrt{m-n} to the power of 2 and get m-n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(m+n-\left(m-n\right))$$

To find the opposite of m-n, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}n}(m+n-m-\left(-n\right))$$

The opposite of -n is n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(m+n-m+n)$$

Combine m and -m to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}n}(n+n)$$

Combine n and n to get 2n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2n)$$

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

$$2n^{1-1}$$

Subtract 1 from 1.

$$2n^{0}$$

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

$$2\times 1$$

For any term t, t\times 1=t and 1t=t.

$$2$$