Solve frac{81x^{14}y^{0}z^{-4}x^{4}z^-4}{81x^12y^-12z^-4}

$$\frac { 81 x ^ { 14 } y ^ { 0 } z ^ { - 4 } x ^ { 4 } z ^ { - 4 } } { 81 x ^ { 12 } y ^ { - 12 } z ^ { - 4 } }$$

Evaluate

$$\frac{x^{6}y^{12}}{z^{4}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 14 and 4 to get 18.

$$\frac{81x^{18}y^{0}z^{-4}z^{-4}}{81x^{12}y^{-12}z^{-4}}$$

To multiply powers of the same base, add their exponents. Add -4 and -4 to get -8.

$$\frac{81x^{18}y^{0}z^{-8}}{81x^{12}y^{-12}z^{-4}}$$

Cancel out 81x^{12} in both numerator and denominator.

$$\frac{z^{-8}y^{0}x^{6}}{y^{-12}z^{-4}}$$

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

$$\frac{z^{-8}x^{6}y^{12}}{z^{-4}}$$

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

$$\frac{x^{6}y^{12}}{z^{4}}$$

Differentiate w.r.t. x

$$\frac{6x^{5}y^{12}}{z^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{81y^{0}x^{4}}{z^{4}z^{4}\times \frac{81}{z^{4}y^{12}}}x^{14-12})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{4}y^{12}}{z^{4}}x^{2})$$

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}.

$$2\times \frac{x^{4}y^{12}}{z^{4}}x^{2-1}$$

Do the arithmetic.

$$\frac{2x^{4}y^{12}}{z^{4}}x^{1}$$

For any term t, t^{1}=t.

$$\frac{2x^{4}y^{12}}{z^{4}}x$$