Solve (4x^2/3xy)div(8x/y^3)

$$( \frac{ 4 { x }^{ 2 } }{ 3xy } ) \div ( \frac{ 8x }{ { y }^{ 3 } } )$$

Evaluate

$$\frac{y^{2}}{6}$$

Solution Steps

Divide \frac{4x^{2}}{3xy} by \frac{8x}{y^{3}} by multiplying \frac{4x^{2}}{3xy} by the reciprocal of \frac{8x}{y^{3}}.

$$\frac{4x^{2}y^{3}}{3xy\times 8x}$$

Cancel out 4xxy in both numerator and denominator.

$$\frac{y^{2}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{y^{2}}{6}$$

Differentiate w.r.t. y

$$\frac{y}{3}$$

Steps Using Definition of a Derivative

Divide \frac{4x^{2}}{3xy} by \frac{8x}{y^{3}} by multiplying \frac{4x^{2}}{3xy} by the reciprocal of \frac{8x}{y^{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4x^{2}y^{3}}{3xy\times 8x})$$

Cancel out 4xxy in both numerator and denominator.

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

Multiply 2 and 3 to get 6.

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

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

$$2\times \frac{1}{6}y^{2-1}$$

Multiply 2 times \frac{1}{6}.

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

Subtract 1 from 2.

$$\frac{1}{3}y^{1}$$

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

$$\frac{1}{3}y$$