Solve (-3x^{13}y^{15})(-3x^{15}y^{18})+(-29x^12y^16)(4x^16y^17)=

$$( - 3 x ^ { 13 } y ^ { 15 } ) ( - 3 x ^ { 15 } y ^ { 18 } ) + ( - 29 x ^ { 12 } y ^ { 16 } ) ( 4 x ^ { 16 } y ^ { 17 } ) =$$

Evaluate

$$-107x^{28}y^{33}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 13 and 15 to get 28.

$$-3x^{28}y^{15}\left(-3\right)y^{18}-29x^{12}y^{16}\times 4x^{16}y^{17}$$

To multiply powers of the same base, add their exponents. Add 15 and 18 to get 33.

$$-3x^{28}y^{33}\left(-3\right)-29x^{12}y^{16}\times 4x^{16}y^{17}$$

To multiply powers of the same base, add their exponents. Add 12 and 16 to get 28.

$$-3x^{28}y^{33}\left(-3\right)-29x^{28}y^{16}\times 4y^{17}$$

To multiply powers of the same base, add their exponents. Add 16 and 17 to get 33.

$$-3x^{28}y^{33}\left(-3\right)-29x^{28}y^{33}\times 4$$

Multiply -3 and -3 to get 9.

$$9x^{28}y^{33}-29x^{28}y^{33}\times 4$$

Multiply -29 and 4 to get -116.

$$9x^{28}y^{33}-116x^{28}y^{33}$$

Combine 9x^{28}y^{33} and -116x^{28}y^{33} to get -107x^{28}y^{33}.

$$-107x^{28}y^{33}$$

Differentiate w.r.t. x

$$-2996x^{27}y^{33}$$

Steps Using Derivative Rule for Sum

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}.

$$13\times 9x^{15}y^{33}x^{13-1}+12\left(-116x^{16}y^{33}\right)x^{12-1}$$

Multiply 13 times -3\left(-3\right)y^{15}x^{15}y^{18}.

$$117x^{15}y^{33}x^{13-1}+12\left(-116x^{16}y^{33}\right)x^{12-1}$$

Subtract 1 from 13.

$$117x^{15}y^{33}x^{12}+12\left(-116x^{16}y^{33}\right)x^{12-1}$$

Multiply 12 times -29\times 4y^{16}x^{16}y^{17}.

$$117x^{15}y^{33}x^{12}+\left(-1392x^{16}y^{33}\right)x^{12-1}$$

Subtract 1 from 12.

$$117x^{15}y^{33}x^{12}+\left(-1392x^{16}y^{33}\right)x^{11}$$