Solve x1/2*x*x1/3

$$x \frac{ 1 }{ 2 } \times x \times x \frac{ 1 }{ 3 }$$

Evaluate

$$\frac{x^{3}}{6}$$

Solution Steps

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

$$x^{2}\times \frac{1}{2}x\times \frac{1}{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{3}\times \frac{1}{2}\times \frac{1}{3}$$

Multiply \frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$x^{3}\times \frac{1\times 1}{2\times 3}$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 3}.

$$x^{3}\times \frac{1}{6}$$

Differentiate w.r.t. x

$$\frac{x^{2}}{2}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}\times \frac{1}{2}x\times \frac{1}{3})$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{1}{2}\times \frac{1}{3})$$

Multiply \frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{1\times 1}{2\times 3})$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}\times \frac{1}{6})$$

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

$$3\times \frac{1}{6}x^{3-1}$$

Multiply 3 times \frac{1}{6}.

$$\frac{1}{2}x^{3-1}$$

Subtract 1 from 3.

$$\frac{1}{2}x^{2}$$