Solve frac{-10x^{6}y^{-2}z^3}{-8x^-3y^-2z^5}

$$\frac { - 10 x ^ { 6 } y ^ { - 2 } z ^ { 3 } } { - 8 x ^ { - 3 } y ^ { - 2 } z ^ { 5 } }$$

Evaluate

$$\frac{5x^{9}}{4z^{2}}$$

Solution Steps

Cancel out 2y^{-2}z^{3} in both numerator and denominator.

$$\frac{-5x^{6}}{-4x^{-3}z^{2}}$$

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

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

Differentiate w.r.t. x

$$\frac{45x^{8}}{4z^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{\frac{10z^{3}}{y^{2}}}{-\frac{8z^{5}}{y^{2}}}\right)x^{6-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{4z^{2}}x^{9})$$

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}.

$$9\times \frac{5}{4z^{2}}x^{9-1}$$

Do the arithmetic.

$$\frac{45}{4z^{2}}x^{8}$$