Solve frac{8x^{2}y^{8}z^-5}{12x^4y^-7z^7}

$$\frac { 8 x ^ { 2 } y ^ { 8 } z ^ { - 5 } } { 12 x ^ { 4 } y ^ { - 7 } z ^ { 7 } }$$

Evaluate

$$\frac{2y^{15}}{3x^{2}z^{12}}$$

Solution Steps

Cancel out 4x^{2} in both numerator and denominator.

$$\frac{2z^{-5}y^{8}}{3y^{-7}x^{2}z^{7}}$$

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

$$\frac{2z^{-5}y^{15}}{3x^{2}z^{7}}$$

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

$$\frac{2y^{15}}{3x^{2}z^{12}}$$

Differentiate w.r.t. x

$$-\frac{4y^{15}}{3x^{3}z^{12}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8y^{8}}{z^{5}\times \frac{12z^{7}}{y^{7}}}x^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{15}}{3z^{12}}x^{-2})$$

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}.

$$-2\times \frac{2y^{15}}{3z^{12}}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{4y^{15}}{3z^{12}}\right)x^{-3}$$