Solve x/1/sqrt{x/x}frac{x/1}{frac{sqrt{x}}{x}}

$$\frac { \frac { x } { 1 } } { \frac { \sqrt { x } } { x } } \frac { \frac { x } { 1 } } { \frac { \sqrt { x } } { x } }$$

Evaluate

$$x^{3}$$

Solution Steps

Multiply \frac{\frac{x}{1}}{\frac{\sqrt{x}}{x}} and \frac{\frac{x}{1}}{\frac{\sqrt{x}}{x}} to get \left(\frac{\frac{x}{1}}{\frac{\sqrt{x}}{x}}\right)^{2}.

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

Divide \frac{x}{1} by \frac{\sqrt{x}}{x} by multiplying \frac{x}{1} by the reciprocal of \frac{\sqrt{x}}{x}.

$$\left(\frac{xx}{\sqrt{x}}\right)^{2}$$

Multiply x and x to get x^{2}.

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

To raise \frac{x^{2}}{\sqrt{x}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{x^{4}}{\left(\sqrt{x}\right)^{2}}$$

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

$$\frac{x^{4}}{x}$$

Cancel out x in both numerator and denominator.

$$x^{3}$$

Differentiate w.r.t. x

$$3x^{2}$$

Steps Using Definition of a Derivative

Multiply \frac{\frac{x}{1}}{\frac{\sqrt{x}}{x}} and \frac{\frac{x}{1}}{\frac{\sqrt{x}}{x}} to get \left(\frac{\frac{x}{1}}{\frac{\sqrt{x}}{x}}\right)^{2}.

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

Divide \frac{x}{1} by \frac{\sqrt{x}}{x} by multiplying \frac{x}{1} by the reciprocal of \frac{\sqrt{x}}{x}.

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

Multiply x and x to get x^{2}.

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

To raise \frac{x^{2}}{\sqrt{x}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{4}}{\left(\sqrt{x}\right)^{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{4}}{x})$$

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3})$$

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

$$3x^{3-1}$$

Subtract 1 from 3.

$$3x^{2}$$