Solve 2x/53y/4x2y/3

$$\frac{ 2x }{ 5 } \frac{ 3y }{ 4x } \frac{ 2y }{ 3 }$$

Evaluate

$$\frac{y^{2}}{5}$$

Solution Steps

Multiply \frac{2x}{5} times \frac{3y}{4x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x\times 3y}{5\times 4x}\times \frac{2y}{3}$$

Cancel out 2x in both numerator and denominator.

$$\frac{3y}{2\times 5}\times \frac{2y}{3}$$

Multiply \frac{3y}{2\times 5} times \frac{2y}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3y\times 2y}{2\times 5\times 3}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{yy}{5}$$

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

$$\frac{y^{2}}{5}$$

Differentiate w.r.t. y

$$\frac{2y}{5}$$

Steps Using Definition of a Derivative

Multiply \frac{2x}{5} times \frac{3y}{4x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{2x\times 3y}{5\times 4x}\times \frac{2y}{3})$$

Cancel out 2x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{3y}{2\times 5}\times \frac{2y}{3})$$

Multiply \frac{3y}{2\times 5} times \frac{2y}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{3y\times 2y}{2\times 5\times 3})$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{yy}{5})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{2}}{5})$$

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

$$2\times \frac{1}{5}y^{2-1}$$

Multiply 2 times \frac{1}{5}.

$$\frac{2}{5}y^{2-1}$$

Subtract 1 from 2.

$$\frac{2}{5}y^{1}$$

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

$$\frac{2}{5}y$$