Solve 3/4x^{4}y^2*(-1/3x^2y^4)div(-12xy^2)

$$\frac { 3 } { 4 } x ^ { 4 } y ^ { 2 } \times ( - \frac { 1 } { 3 } x ^ { 2 } y ^ { 4 } ) \div ( - 12 x y ^ { 2 } )$$

Evaluate

$$\frac{y^{4}x^{5}}{48}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\frac{3}{4}x^{6}y^{2}\left(-\frac{1}{3}\right)y^{4}}{-12xy^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\frac{3}{4}x^{6}y^{6}\left(-\frac{1}{3}\right)}{-12xy^{2}}$$

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

$$\frac{-\frac{1}{3}\times \frac{3}{4}y^{4}x^{5}}{-12}$$

Multiply -\frac{1}{3} and \frac{3}{4} to get -\frac{1}{4}.

$$\frac{-\frac{1}{4}y^{4}x^{5}}{-12}$$

Divide -\frac{1}{4}y^{4}x^{5} by -12 to get \frac{1}{48}y^{4}x^{5}.

$$\frac{1}{48}y^{4}x^{5}$$

Differentiate w.r.t. x

$$\frac{5\left(xy\right)^{4}}{48}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{\frac{\frac{3}{4}x^{2}y^{2}y^{4}}{3}}{-12y^{2}}\right)x^{4-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}y^{4}}{48}x^{3})$$

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}.

$$3\times \frac{x^{2}y^{4}}{48}x^{3-1}$$

Do the arithmetic.

$$\frac{x^{2}y^{4}}{16}x^{2}$$