Solve frac{3x^{2}y^{-3}z^2*4z^3*2xz}{2x^4y^4}

$$\frac { 3 x ^ { 2 } y ^ { - 3 } z ^ { 2 } \cdot 4 z ^ { 3 } \cdot 2 x z } { 2 x ^ { 4 } y ^ { 4 } }$$

Evaluate

$$\frac{12z^{6}}{xy^{7}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{3x^{3}y^{-3}z^{2}\times 4z^{3}\times 2z}{2x^{4}y^{4}}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

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

To multiply powers of the same base, add their exponents. Add 5 and 1 to get 6.

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

Cancel out 2x^{3} in both numerator and denominator.

$$\frac{2\times 2\times 3y^{-3}z^{6}}{xy^{4}}$$

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

$$\frac{2\times 2\times 3z^{6}}{xy^{7}}$$

Multiply 2 and 2 to get 4.

$$\frac{4\times 3z^{6}}{xy^{7}}$$

Multiply 4 and 3 to get 12.

$$\frac{12z^{6}}{xy^{7}}$$

Differentiate w.r.t. x

$$-\frac{12z^{6}}{x^{2}y^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{24xz^{6}}{y^{3}\times 2y^{4}}x^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12xz^{6}}{y^{7}}x^{-2})$$

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}.

$$-2\times \frac{12xz^{6}}{y^{7}}x^{-2-1}$$

Do the arithmetic.

$$\left(-\frac{24xz^{6}}{y^{7}}\right)x^{-3}$$