Solve frac{x^{10}y^{-8}z^3}{x^7y^2z^-3}

$$\frac { x ^ { 10 } y ^ { - 8 } z ^ { 3 } } { x ^ { 7 } y ^ { 2 } z ^ { - 3 } }$$

Evaluate

$$\frac{x^{3}z^{6}}{y^{10}}$$

Solution Steps

Cancel out x^{7} in both numerator and denominator.

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

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

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

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

$$\frac{x^{3}z^{6}}{y^{10}}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{z^{3}}{y^{8}\times \frac{y^{2}}{z^{3}}}x^{10-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{z^{6}}{y^{10}}x^{3})$$

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}.

$$3\times \frac{z^{6}}{y^{10}}x^{3-1}$$

Do the arithmetic.

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