Solve 49x^2yz^3/(7xz)(3x^2y^2)

$$\frac { 49 x ^ { 2 } y z ^ { 3 } } { ( 7 x z ) ( 3 x ^ { 2 } y ^ { 2 } ) }$$

Evaluate

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

Solution Steps

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

$$\frac{49x^{2}yz^{3}}{7x^{3}z\times 3y^{2}}$$

Cancel out 7yzx^{2} in both numerator and denominator.

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

Differentiate w.r.t. x

$$-\frac{7\times \left(\frac{z}{x}\right)^{2}}{3y}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{49yz^{3}}{21zx^{2}y^{2}}x^{2-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7z^{2}}{3yx^{2}}x^{1})$$

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}.

$$\frac{7z^{2}}{3yx^{2}}x^{1-1}$$

Do the arithmetic.

$$\frac{7z^{2}}{3yx^{2}}x^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{7z^{2}}{3yx^{2}}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{7z^{2}}{3yx^{2}}$$