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

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

Evaluate

$$\frac{y^{12}}{\left(xz\right)^{4}}$$

Solution Steps

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

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

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

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

Cancel out 81x^{8} in both numerator and denominator.

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

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

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

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

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

$$-8\times \frac{x^{4}y^{12}}{z^{4}}x^{-8-1}$$

Do the arithmetic.

$$\left(-\frac{8x^{4}y^{12}}{z^{4}}\right)x^{-9}$$