Solve frac{x^{-2}y^{-5}z^-1}{x^4y^6z^7}

$$\frac { x ^ { - 2 } y ^ { - 5 } z ^ { - 1 } } { x ^ { 4 } y ^ { 6 } z ^ { 7 } }$$

Differentiate w.r.t. x

$$-\frac{6}{x^{7}z^{8}y^{11}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{zy^{5}y^{6}z^{7}}x^{-2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{z^{8}y^{11}}x^{-6})$$

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}.

$$-6\times \frac{1}{z^{8}y^{11}}x^{-6-1}$$

Do the arithmetic.

$$\left(-\frac{6}{z^{8}y^{11}}\right)x^{-7}$$

Evaluate

$$\frac{1}{x^{6}z^{8}y^{11}}$$

Solution Steps

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

$$\frac{1}{x^{6}z^{8}y^{11}}$$