Solve frac{(12x^{3}y^{2})*(3x^{5}yz^6)}{9x^-4y^3z^3}=

$$\frac { ( 12 x ^ { 3 } y ^ { 2 } ) \times ( 3 x ^ { 5 } y z ^ { 6 } ) } { 9 x ^ { - 4 } y ^ { 3 } z ^ { 3 } } =$$

Evaluate

$$4z^{3}x^{12}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$\frac{12x^{8}y^{2}\times 3yz^{6}}{9x^{-4}y^{3}z^{3}}$$

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

$$\frac{12x^{8}y^{3}\times 3z^{6}}{9x^{-4}y^{3}z^{3}}$$

Cancel out 3\times 3y^{3}z^{3} in both numerator and denominator.

$$\frac{4z^{3}x^{8}}{x^{-4}}$$

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

$$4z^{3}x^{12}$$

Differentiate w.r.t. x

$$48z^{3}x^{11}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{36y^{3}x^{5}z^{6}}{9y^{3}z^{3}}x^{3-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4z^{3}x^{5}x^{7})$$

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}.

$$7\times 4z^{3}x^{5}x^{7-1}$$

Do the arithmetic.

$$28z^{3}x^{5}x^{6}$$