Solve frac{4x^{2}y^3*9xy^2}{12x^4y^2}=

$$\frac { 4 x ^ { 2 } y ^ { 3 } \times 9 x y ^ { 2 } } { 12 x ^ { 4 } y ^ { 2 } } =$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{4x^{3}y^{3}\times 9y^{2}}{12x^{4}y^{2}}$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{4x^{3}y^{5}\times 9}{12x^{4}y^{2}}$$

Cancel out 3\times 4y^{2}x^{3} in both numerator and denominator.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{36xy^{5}}{12y^{2}}x^{2-4})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-6xy^{3}\right)x^{-3}$$