Solve (x^3y+3x^2y^2+2xy^5)+(2)

$$( x ^ { 3 } y + 3 x ^ { 2 } y ^ { 2 } + 2 x y ^ { 5 } ) + ( 2 )$$

Differentiate w.r.t. x

$$y\left(3x^{2}+6xy+2y^{4}\right)$$

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}.

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

Subtract 1 from 3.

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

Multiply 2 times 3y^{2}.

$$3yx^{2}+6y^{2}x^{2-1}+2y^{5}x^{1-1}$$

Subtract 1 from 2.

$$3yx^{2}+6y^{2}x^{1}+2y^{5}x^{1-1}$$

Subtract 1 from 1.

$$3yx^{2}+6y^{2}x^{1}+2y^{5}x^{0}$$

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

$$3yx^{2}+6y^{2}x+2y^{5}x^{0}$$

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

$$3yx^{2}+6y^{2}x+2y^{5}\times 1$$

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

$$3yx^{2}+6y^{2}x+2y^{5}$$

Evaluate

$$2xy^{5}+yx^{3}+3\left(xy\right)^{2}+2$$