Solve (6/5x^{5}y^{4}z^8):(5/3x^3y^3z^7)=

$$( \frac { 6 } { 5 } x ^ { 5 } y ^ { 4 } z ^ { 8 } ) : ( \frac { 5 } { 3 } x ^ { 3 } y ^ { 3 } z ^ { 7 } ) =$$

Evaluate

$$\frac{18yzx^{2}}{25}$$

Solution Steps

Cancel out x^{3}y^{3}z^{7} in both numerator and denominator.

$$\frac{\frac{6}{5}yzx^{2}}{\frac{5}{3}}$$

Divide \frac{6}{5}yzx^{2} by \frac{5}{3} by multiplying \frac{6}{5}yzx^{2} by the reciprocal of \frac{5}{3}.

$$\frac{\frac{6}{5}yzx^{2}\times 3}{5}$$

Multiply \frac{6}{5} and 3 to get \frac{18}{5}.

$$\frac{\frac{18}{5}yzx^{2}}{5}$$

Divide \frac{18}{5}yzx^{2} by 5 to get \frac{18}{25}yzx^{2}.

$$\frac{18}{25}yzx^{2}$$

Differentiate w.r.t. x

$$\frac{36xyz}{25}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{4}z^{8}}{5\times \frac{5y^{3}z^{7}}{3}}x^{5-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{18yz}{25}x^{2})$$

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}.

$$2\times \frac{18yz}{25}x^{2-1}$$

Do the arithmetic.

$$\frac{36yz}{25}x^{1}$$

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

$$\frac{36yz}{25}x$$