Solve frac{(2x^{3}y^7)(-12x^5y^9)}{-8xy^12}

$$\frac { ( 2 x ^ { 3 } y ^ { 7 } ) ( - 12 x ^ { 5 } y ^ { 9 } ) } { - 8 x y ^ { 12 } }$$

Evaluate

$$3y^{4}x^{7}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$\frac{2x^{8}y^{7}\left(-12\right)y^{9}}{-8xy^{12}}$$

To multiply powers of the same base, add their exponents. Add 7 and 9 to get 16.

$$\frac{2x^{8}y^{16}\left(-12\right)}{-8xy^{12}}$$

Cancel out 2\times 4xy^{12} in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

$$3y^{4}x^{7}$$

Differentiate w.r.t. x

$$21y^{4}x^{6}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{24x^{5}y^{16}}{-8y^{12}}\right)x^{3-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3y^{4}x^{5}x^{2})$$

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}.

$$2\times 3y^{4}x^{5}x^{2-1}$$

Do the arithmetic.

$$6y^{4}x^{5}x^{1}$$

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

$$6y^{4}x^{5}x$$