Solve frac{sqrt{n}+1-sqrt{n}}{sqrt{n}(sqrt{n}*1}

$$\frac { \sqrt { n } + 1 - \sqrt { n } } { \sqrt { n } ( \sqrt { n } * 1 }$$

Evaluate

$$\frac{1}{n}$$

Solution Steps

Multiply \sqrt{n} and \sqrt{n} to get \left(\sqrt{n}\right)^{2}.

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

Combine \sqrt{n} and -\sqrt{n} to get 0.

$$\frac{1}{1\left(\sqrt{n}\right)^{2}}$$

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

$$\frac{1}{1n}$$

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

$$\frac{1}{n}$$

Differentiate w.r.t. n

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

Steps Using Definition of a Derivative

Multiply \sqrt{n} and \sqrt{n} to get \left(\sqrt{n}\right)^{2}.

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

Combine \sqrt{n} and -\sqrt{n} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{1\left(\sqrt{n}\right)^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{1n})$$

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

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

Subtract 1 from -1.

$$-n^{-2}$$