Solve frac{x^{3}y^{3}}{3x^-1y^4*4x^4y^-3}

$$\frac { x ^ { 3 } y ^ { 3 } } { 3 x ^ { - 1 } y ^ { 4 } \cdot 4 x ^ { 4 } y ^ { - 3 } }$$

Evaluate

$$\frac{y^{2}}{12}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$\frac{x^{3}y^{3}}{3x^{3}y^{4}\times 4y^{-3}}$$

To multiply powers of the same base, add their exponents. Add 4 and -3 to get 1.

$$\frac{x^{3}y^{3}}{3x^{3}y^{1}\times 4}$$

Cancel out yx^{3} in both numerator and denominator.

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

Multiply 3 and 4 to get 12.

$$\frac{y^{2}}{12}$$

Differentiate w.r.t. y

$$\frac{y}{6}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -1 and 4 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x^{3}y^{3}}{3x^{3}y^{4}\times 4y^{-3}})$$

To multiply powers of the same base, add their exponents. Add 4 and -3 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{x^{3}y^{3}}{3x^{3}y^{1}\times 4})$$

Cancel out yx^{3} in both numerator and denominator.

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

Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{2}}{12})$$

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

$$2\times \frac{1}{12}y^{2-1}$$

Multiply 2 times \frac{1}{12}.

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

Subtract 1 from 2.

$$\frac{1}{6}y^{1}$$

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

$$\frac{1}{6}y$$