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

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

Evaluate

$$\frac{y^{9}}{3}$$

Solution Steps

Cancel out x in both numerator and denominator.

$$\frac{x^{2}y^{5}}{3}\times \left(\frac{y^{4}}{x^{2}}\right)$$

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

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

Cancel out x^{2} in both numerator and denominator.

$$\frac{y^{4}y^{5}}{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{y^{9}}{3}$$

Differentiate w.r.t. y

$$3y^{8}$$

Steps Using Definition of a Derivative

Cancel out x in both numerator and denominator.

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

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

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

Cancel out x^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{4}y^{5}}{3})$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{9}}{3})$$

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

$$9\times \left(\frac{1}{3}\right)y^{9-1}$$

Multiply 9 times \frac{1}{3}.

$$3y^{9-1}$$

Subtract 1 from 9.

$$3y^{8}$$