Solve (frac{20x^{-9}y^{2}z^5}{15x^-4y^-2z^-8})

$$( \frac { 20 x ^ { - 9 } y ^ { 2 } z ^ { 5 } } { 15 x ^ { - 4 } y ^ { - 2 } z ^ { - 8 } } )$$

Evaluate

$$\frac{4y^{4}z^{13}}{3x^{5}}$$

Solution Steps

Cancel out 5 in both numerator and denominator.

$$\frac{4x^{-9}y^{2}z^{5}}{3z^{-8}x^{-4}y^{-2}}$$

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

$$\frac{4x^{-9}y^{4}z^{13}}{3x^{-4}}$$

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

$$\frac{4y^{4}z^{13}}{3x^{5}}$$

Differentiate w.r.t. x

$$-\frac{20y^{4}z^{13}}{3x^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20y^{2}z^{5}y^{2}z^{8}}{15}x^{-9-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{4}z^{13}}{3}x^{-5})$$

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}.

$$-5\times \frac{4y^{4}z^{13}}{3}x^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{20y^{4}z^{13}}{3}\right)x^{-6}$$