Solve 25y/2x*4xy/5y^3

$$\frac { 25 y } { 2 x } \cdot \frac { 4 x y } { 5 y ^ { 3 } }$$

Evaluate

$$\frac{10}{y}$$

Solution Steps

Cancel out y in both numerator and denominator.

$$\frac{25y}{2x}\times \frac{4x}{5y^{2}}$$

Multiply \frac{25y}{2x} times \frac{4x}{5y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{25y\times 4x}{2x\times 5y^{2}}$$

Cancel out 2\times 5xy in both numerator and denominator.

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

Multiply 2 and 5 to get 10.

$$\frac{10}{y}$$

Differentiate w.r.t. y

$$-\frac{10}{y^{2}}$$

Steps Using Definition of a Derivative

Cancel out y in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{25y}{2x}\times \frac{4x}{5y^{2}})$$

Multiply \frac{25y}{2x} times \frac{4x}{5y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{25y\times 4x}{2x\times 5y^{2}})$$

Cancel out 2\times 5xy in both numerator and denominator.

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

Multiply 2 and 5 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{10}{y})$$

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

$$-10y^{-1-1}$$

Subtract 1 from -1.

$$-10y^{-2}$$