Solve frac{2x^{5}y^{4}*2xy^{3}}{2x^2y^5*3x^2y^3}

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

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 1 to get 6.

$$\frac{2x^{6}y^{4}\times 2y^{3}}{2x^{2}y^{5}\times 3x^{2}y^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{2x^{6}y^{7}\times 2}{2x^{2}y^{5}\times 3x^{2}y^{3}}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{2x^{6}y^{7}\times 2}{2x^{4}y^{5}\times 3y^{3}}$$

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$\frac{2x^{6}y^{7}\times 2}{2x^{4}y^{8}\times 3}$$

Cancel out 2x^{4}y^{7} in both numerator and denominator.

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

Differentiate w.r.t. x

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

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4xy^{7}}{6x^{2}y^{8}}x^{5-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3xy}x^{3})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3\times \frac{2}{3xy}x^{3-1}$$

Do the arithmetic.

$$\frac{2}{xy}x^{2}$$