Solve (frac{63x^{-3}y^{4}z^-5}{35x^8y^-7z^-4})

$$( \frac { 63 x ^ { - 3 } y ^ { 4 } z ^ { - 5 } } { 35 x ^ { 8 } y ^ { - 7 } z ^ { - 4 } } )$$

Evaluate

$$\frac{9\times \left(\frac{y}{x}\right)^{11}}{5z}$$

Solution Steps

Cancel out 7 in both numerator and denominator.

$$\frac{9z^{-5}x^{-3}y^{4}}{5y^{-7}z^{-4}x^{8}}$$

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

$$\frac{9z^{-5}x^{-3}y^{11}}{5z^{-4}x^{8}}$$

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

$$\frac{9y^{11}}{5z^{1}x^{11}}$$

Calculate z to the power of 1 and get z.

$$\frac{9y^{11}}{5zx^{11}}$$

Differentiate w.r.t. x

$$-\frac{99y^{11}}{5zx^{12}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{63y^{4}}{z^{5}\times \frac{35}{z^{4}y^{7}}}x^{-3-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9y^{11}}{5z}x^{-11})$$

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}.

$$-11\times \frac{9y^{11}}{5z}x^{-11-1}$$

Do the arithmetic.

$$\left(-\frac{99y^{11}}{5z}\right)x^{-12}$$