Solve 24x^-2y^3/12x^-3y^5

$$\frac { 24 x ^ { - 2 } y ^ { 3 } } { 12 x ^ { - 3 } y ^ { 5 } }$$

Evaluate

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

Solution Steps

Cancel out 12y^{3} in both numerator and denominator.

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

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

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

Calculate x to the power of 1 and get x.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{24y^{3}}{12y^{5}}x^{-2-\left(-3\right)})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\frac{2}{y^{2}}x^{0}$$

For any term t except 0, t^{0}=1.

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

For any term t, t\times 1=t and 1t=t.

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