Solve frac{3x^{2}y*18y^-1x^-1}{9x^-4y^-3}

$$\frac { 3 x ^ { 2 } y \times 18 y ^ { - 1 } x ^ { - 1 } } { 9 x ^ { - 4 } y ^ { - 3 } }$$

Evaluate

$$6y^{3}x^{5}$$

Solution Steps

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

$$\frac{3x^{1}y\times 18y^{-1}}{9x^{-4}y^{-3}}$$

Multiply y and y^{-1} to get 1.

$$\frac{3x^{1}\times 18}{9x^{-4}y^{-3}}$$

Cancel out 3\times 3 in both numerator and denominator.

$$\frac{6x^{1}}{x^{-4}y^{-3}}$$

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

$$\frac{6x^{5}}{y^{-3}}$$

Differentiate w.r.t. x

$$30y^{3}x^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{54y}{xy\times \frac{9}{y^{3}}}x^{2-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{3}}{x}x^{6})$$

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}.

$$6\times \frac{6y^{3}}{x}x^{6-1}$$

Do the arithmetic.

$$\frac{36y^{3}}{x}x^{5}$$