Solve (3+sqrt{2})y(3-sqrt{2})

$$( 3 + \sqrt { 2 } ) y ( 3 - \sqrt { 2 } )$$

Evaluate

$$7y$$

Solution Steps

Use the distributive property to multiply 3+\sqrt{2} by y.

$$\left(3y+\sqrt{2}y\right)\left(3-\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 3y+\sqrt{2}y by each term of 3-\sqrt{2}.

$$9y-3\sqrt{2}y+3\sqrt{2}y-y\left(\sqrt{2}\right)^{2}$$

Combine -3\sqrt{2}y and 3\sqrt{2}y to get 0.

$$9y-y\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9y-y\times 2$$

Multiply -1 and 2 to get -2.

$$9y-2y$$

Combine 9y and -2y to get 7y.

$$7y$$

Differentiate w.r.t. y

$$7$$

Steps Using Definition of a Derivative

Use the distributive property to multiply 3+\sqrt{2} by y.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(3y+\sqrt{2}y\right)\left(3-\sqrt{2}\right))$$

Apply the distributive property by multiplying each term of 3y+\sqrt{2}y by each term of 3-\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(9y-3\sqrt{2}y+3\sqrt{2}y-y\left(\sqrt{2}\right)^{2})$$

Combine -3\sqrt{2}y and 3\sqrt{2}y to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}y}(9y-y\left(\sqrt{2}\right)^{2})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(9y-y\times 2)$$

Multiply -1 and 2 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(9y-2y)$$

Combine 9y and -2y to get 7y.

$$\frac{\mathrm{d}}{\mathrm{d}y}(7y)$$

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

$$7y^{1-1}$$

Subtract 1 from 1.

$$7y^{0}$$

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

$$7\times 1$$

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

$$7$$