Solve 2x^{6}y^2*x^3y+(-25x^8y^2)*(-xy)

$$2 x ^ { 6 } y ^ { 2 } \cdot x ^ { 3 } y + ( - 25 x ^ { 8 } y ^ { 2 } ) \cdot ( - x y )$$

Evaluate

$$27y^{3}x^{9}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 6 and 3 to get 9.

$$2x^{9}y^{2}y-25x^{8}y^{2}\left(-x\right)y$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$2x^{9}y^{3}-25x^{8}y^{2}\left(-x\right)y$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$2x^{9}y^{3}-25x^{8}y^{3}\left(-x\right)$$

Multiply -25 and -1 to get 25.

$$2x^{9}y^{3}+25x^{8}y^{3}x$$

To multiply powers of the same base, add their exponents. Add 8 and 1 to get 9.

$$2x^{9}y^{3}+25x^{9}y^{3}$$

Combine 2x^{9}y^{3} and 25x^{9}y^{3} to get 27x^{9}y^{3}.

$$27x^{9}y^{3}$$

Differentiate w.r.t. x

$$243y^{3}x^{8}$$

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}.

$$6\times 2yy^{2}x^{3}x^{6-1}+8\times 25xy^{3}x^{8-1}$$

Multiply 6 times 2y^{2}x^{3}y.

$$12x^{3}y^{3}x^{6-1}+8\times 25xy^{3}x^{8-1}$$

Subtract 1 from 6.

$$12x^{3}y^{3}x^{5}+8\times 25xy^{3}x^{8-1}$$

Multiply 8 times -25y^{2}\left(-1\right)xy.

$$12x^{3}y^{3}x^{5}+200xy^{3}x^{8-1}$$

Subtract 1 from 8.

$$12x^{3}y^{3}x^{5}+200xy^{3}x^{7}$$