Solve 1/x*x^3/3

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

Evaluate

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

Solution Steps

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

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

Cancel out x in both numerator and denominator.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Product

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

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

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}.

$$\frac{1}{x}\times 3\times \frac{1}{3}x^{3-1}+\frac{1}{3}x^{3}\left(-1\right)x^{-1-1}$$

Simplify.

$$\frac{1}{x}x^{2}+\frac{1}{3}x^{3}\left(-1\right)x^{-2}$$

To multiply powers of the same base, add their exponents.

$$x^{-1+2}+\frac{1}{3}\left(-1\right)x^{3-2}$$

Simplify.

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

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

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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

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

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