Solve frac{48x^{3}y^5z^9}{-8x^2z^4}

$$\frac { 48 x ^ { 3 } y ^ { 5 } z ^ { 9 } } { - 8 x ^ { 2 } z ^ { 4 } }$$

Evaluate

$$-6x\left(yz\right)^{5}$$

Solution Steps

Cancel out 8x^{2}z^{4} in both numerator and denominator.

$$\frac{6xy^{5}z^{5}}{-1}$$

Anything divided by -1 gives its opposite.

$$-6xy^{5}z^{5}$$

Differentiate w.r.t. x

$$-6\left(yz\right)^{5}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{48y^{5}z^{9}}{-8z^{4}}x^{3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-6y^{5}z^{5}\right)x^{1})$$

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}.

$$\left(-6y^{5}z^{5}\right)x^{1-1}$$

Do the arithmetic.

$$\left(-6y^{5}z^{5}\right)x^{0}$$

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

$$\left(-6y^{5}z^{5}\right)\times 1$$

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

$$-6y^{5}z^{5}$$