Solve frac{24x^{-3}y^{4}z^2}{36x^-2y^-7z^5}

$$\frac { 24 x ^ { - 3 } y ^ { 4 } z ^ { 2 } } { 36 x ^ { - 2 } y ^ { - 7 } z ^ { 5 } }$$

Evaluate

$$\frac{2y^{11}}{3xz^{3}}$$

Solution Steps

Cancel out 12z^{2} in both numerator and denominator.

$$\frac{2x^{-3}y^{4}}{3y^{-7}x^{-2}z^{3}}$$

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

$$\frac{2x^{-3}y^{11}}{3x^{-2}z^{3}}$$

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

$$\frac{2y^{11}}{3x^{1}z^{3}}$$

Calculate x to the power of 1 and get x.

$$\frac{2y^{11}}{3xz^{3}}$$

Differentiate w.r.t. x

$$-\frac{2y^{11}}{3x^{2}z^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{24z^{2}y^{4}y^{7}}{36z^{5}}x^{-3-\left(-2\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{11}}{3z^{3}}\times \frac{1}{x})$$

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}.

$$-\frac{2y^{11}}{3z^{3}}x^{-1-1}$$

Do the arithmetic.

$$\left(-\frac{2y^{11}}{3z^{3}}\right)x^{-2}$$