Solve frac{21x^{3}y^{-5}z^-2}{35x^-1y^2z^-7}

$$\frac { 21 x ^ { 3 } y ^ { - 5 } z ^ { - 2 } } { 35 x ^ { - 1 } y ^ { 2 } z ^ { - 7 } }$$

Evaluate

$$\frac{3x^{4}z^{5}}{5y^{7}}$$

Solution Steps

Cancel out 7 in both numerator and denominator.

$$\frac{3y^{-5}z^{-2}x^{3}}{5z^{-7}\times \frac{1}{x}y^{2}}$$

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

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

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

$$\frac{3x^{4}z^{5}}{5y^{7}}$$

Differentiate w.r.t. x

$$\frac{12x^{3}z^{5}}{5y^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{21}{z^{2}y^{5}\times \frac{35y^{2}}{z^{7}}}x^{3-\left(-1\right)})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\frac{12z^{5}}{5y^{7}}x^{3}$$