Solve frac{-14x^{7}y^{5}y^{4}z^2}{-7x^3yw^2z^3}

$$\frac { - 14 x ^ { 7 } y ^ { 5 } y ^ { 4 } z ^ { 2 } } { - 7 x ^ { 3 } y w ^ { 2 } z ^ { 3 } }$$

Evaluate

$$\frac{2x^{4}y^{8}}{zw^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 4 to get 9.

$$\frac{-14x^{7}y^{9}z^{2}}{-7x^{3}yw^{2}z^{3}}$$

Cancel out 7yz^{2}x^{3} in both numerator and denominator.

$$\frac{-2x^{4}y^{8}}{-zw^{2}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{2x^{4}y^{8}}{zw^{2}}$$

Differentiate w.r.t. x

$$\frac{8x^{3}y^{8}}{zw^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{14z^{2}y^{4}y^{5}}{-7yw^{2}z^{3}}\right)x^{7-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2y^{8}}{zw^{2}}x^{4})$$

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}.

$$4\times \frac{2y^{8}}{zw^{2}}x^{4-1}$$

Do the arithmetic.

$$\frac{8y^{8}}{zw^{2}}x^{3}$$