Solve 3x^-2y^-3/6x^-4y^-1

$$\frac { 3 x ^ { - 2 } y ^ { - 3 } } { 6 x ^ { - 4 } y ^ { - 1 } }$$

Evaluate

$$\frac{\left(\frac{x}{y}\right)^{2}}{2}$$

Solution Steps

Cancel out 3 in both numerator and denominator.

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

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

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

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

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{y^{3}\times \frac{6}{y}}x^{-2-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2y^{2}}x^{2})$$

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}.

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

Do the arithmetic.

$$\frac{1}{y^{2}}x^{1}$$

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

$$\frac{1}{y^{2}}x$$