Solve frac{2x^{-2}y^{4}z^{2}*3x^-1z^3}{4x^3z^3}

$$\frac { 2 x ^ { - 2 } y ^ { 4 } z ^ { 2 } \cdot 3 x ^ { - 1 } z ^ { 3 } } { 4 x ^ { 3 } z ^ { 3 } }$$

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

Differentiate w.r.t. x

$$-\frac{9z^{2}y^{4}}{x^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{4}z^{5}}{x\times 4z^{3}}x^{-2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3z^{2}y^{4}}{2x}x^{-5})$$

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}.

$$-5\times \frac{3z^{2}y^{4}}{2x}x^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{15z^{2}y^{4}}{2x}\right)x^{-6}$$