Solve x^2*x*x^3/xfrac{1{2}*x*x1/3}

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

Evaluate

$$6x^{3}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{x^{6}}{x\times \frac{1}{2}xx\times \frac{1}{3}}$$

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

$$\frac{x^{6}}{x^{2}\times \frac{1}{2}x\times \frac{1}{3}}$$

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

$$\frac{x^{6}}{x^{3}\times \frac{1}{2}\times \frac{1}{3}}$$

Cancel out x^{3} in both numerator and denominator.

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

Multiply \frac{1}{3} and \frac{1}{2} to get \frac{1}{6}.

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

Divide x^{3} by \frac{1}{6} by multiplying x^{3} by the reciprocal of \frac{1}{6}.

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

Differentiate w.r.t. x

$$18x^{2}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{2}x^{1})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$6x^{2}x^{1-1}$$

Do the arithmetic.

$$6x^{2}x^{0}$$

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

$$6x^{2}\times 1$$

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

$$6x^{2}$$