Solve 4x^{7}y^6+9x^3y*2x^4y^5

$$4 x ^ { 7 } y ^ { 6 } + 9 x ^ { 3 } y \cdot 2 x ^ { 4 } y ^ { 5 }$$

Evaluate

$$22y^{6}x^{7}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 4 to get 7.

$$4x^{7}y^{6}+9x^{7}y\times 2y^{5}$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$4x^{7}y^{6}+9x^{7}y^{6}\times 2$$

Multiply 9 and 2 to get 18.

$$4x^{7}y^{6}+18x^{7}y^{6}$$

Combine 4x^{7}y^{6} and 18x^{7}y^{6} to get 22x^{7}y^{6}.

$$22x^{7}y^{6}$$

Differentiate w.r.t. x

$$154\left(xy\right)^{6}$$

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}.

$$7\times 4y^{6}x^{7-1}+3\times 18x^{4}y^{6}x^{3-1}$$

Multiply 7 times 4y^{6}.

$$28y^{6}x^{7-1}+3\times 18x^{4}y^{6}x^{3-1}$$

Subtract 1 from 7.

$$28y^{6}x^{6}+3\times 18x^{4}y^{6}x^{3-1}$$

Multiply 3 times 2\times 9yx^{4}y^{5}.

$$28y^{6}x^{6}+54x^{4}y^{6}x^{3-1}$$

Subtract 1 from 3.

$$28y^{6}x^{6}+54x^{4}y^{6}x^{2}$$