Solve frac{12x^{3}y^9z^-7}{2x^-4y^-6}

$$\frac { 12 x ^ { 3 } y ^ { 9 } z ^ { - 7 } } { 2 x ^ { - 4 } y ^ { - 6 } }$$

Evaluate

$$6\times \left(\frac{x}{z}\right)^{7}y^{15}$$

Solution Steps

Cancel out 2 in both numerator and denominator.

$$\frac{6z^{-7}x^{3}y^{9}}{y^{-6}x^{-4}}$$

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

$$6z^{-7}x^{7}y^{15}$$

Differentiate w.r.t. x

$$\frac{42x^{6}y^{15}}{z^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12y^{9}}{z^{7}\times \frac{2}{y^{6}}}x^{3-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{15}}{z^{7}}x^{7})$$

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}.

$$7\times \frac{6y^{15}}{z^{7}}x^{7-1}$$

Do the arithmetic.

$$\frac{42y^{15}}{z^{7}}x^{6}$$