Solve -x^5+(y^2+x^2y^2)+({x}^5+{y}^2-{x}^2{y}^2)

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

Evaluate

$$2y^{2}$$

Solution Steps

Combine -x^{5} and x^{5} to get 0.

$$y^{2}+x^{2}y^{2}+y^{2}-x^{2}y^{2}$$

Combine y^{2} and y^{2} to get 2y^{2}.

$$2y^{2}+x^{2}y^{2}-x^{2}y^{2}$$

Combine x^{2}y^{2} and -x^{2}y^{2} to get 0.

$$2y^{2}$$

Differentiate w.r.t. y

$$4y$$

Steps Using Definition of a Derivative

Combine -x^{5} and x^{5} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{2}+x^{2}y^{2}+y^{2}-x^{2}y^{2})$$

Combine y^{2} and y^{2} to get 2y^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{2}+x^{2}y^{2}-x^{2}y^{2})$$

Combine x^{2}y^{2} and -x^{2}y^{2} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{2})$$

The derivative of ax^{n} is nax^{n-1}.

$$2\times 2y^{2-1}$$

Multiply 2 times 2.

$$4y^{2-1}$$

Subtract 1 from 2.

$$4y^{1}$$

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

$$4y$$