Solve frac{[x^{2}y^{-4}(-z^3)}{x^6y^-3z^7}

$$\frac { [ x ^ { 2 } y ^ { - 4 } ( - z ^ { 3 } ) } { x ^ { 6 } y ^ { - 3 } z ^ { 7 } }$$

Evaluate

$$-\frac{1}{y\left(xz\right)^{4}}$$

Solution Steps

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

$$\frac{y^{-4}\left(-z^{3}\right)}{y^{-3}x^{4}z^{7}}$$

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

$$\frac{-z^{3}}{y^{1}x^{4}z^{7}}$$

Calculate y to the power of 1 and get y.

$$\frac{-z^{3}}{yx^{4}z^{7}}$$

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

$$\frac{-1}{yx^{4}z^{4}}$$

Differentiate w.r.t. x

$$\frac{4}{yz^{4}x^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(-\frac{z^{3}}{y^{4}}\right)y^{3}}{z^{7}}x^{2-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{1}{yz^{4}}\right)x^{-4})$$

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}.

$$-4\left(-\frac{1}{yz^{4}}\right)x^{-4-1}$$

Do the arithmetic.

$$\frac{4}{yz^{4}}x^{-5}$$