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

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

Evaluate

$$n$$

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}.

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

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

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

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

$$x+n-x$$

Combine x and -x to get 0.

$$n$$

Differentiate w.r.t. n

$$1$$

Steps Using Definition of a Derivative

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

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

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

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

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

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

Combine x and -x to get 0.

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

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

$$n^{1-1}$$

Subtract 1 from 1.

$$n^{0}$$

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

$$1$$