Solve frac{8x^{-5}y^4z^-8}{4x^9y^-6}

$$\frac { 8 x ^ { - 5 } y ^ { 4 } z ^ { - 8 } } { 4 x ^ { 9 } y ^ { - 6 } }$$

Evaluate

$$\frac{2y^{10}}{z^{8}x^{14}}$$

Solution Steps

Cancel out 4 in both numerator and denominator.

$$\frac{2z^{-8}x^{-5}y^{4}}{y^{-6}x^{9}}$$

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

$$\frac{2z^{-8}x^{-5}y^{10}}{x^{9}}$$

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

$$\frac{2z^{-8}y^{10}}{x^{14}}$$

Differentiate w.r.t. x

$$-\frac{28y^{10}}{z^{8}x^{15}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8y^{4}}{z^{8}\times \frac{4}{y^{6}}}x^{-5-9})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{10}}{z^{8}}x^{-14})$$

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}.

$$-14\times \frac{2y^{10}}{z^{8}}x^{-14-1}$$

Do the arithmetic.

$$\left(-\frac{28y^{10}}{z^{8}}\right)x^{-15}$$