#### To determine

**To find:** The derivative dydx by implicit differentiation.

#### Answer

The implicit differentiation of 2x2+x+xy=1 is dydx=−(4x+1+y)x.

#### Explanation

**Given:**

The equation 2x2+x+xy=1.

**Derivative rules:**

(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

(2) Power rule: ddx(xn)=nxn−1

(3) Difference rule: ddx(f−g)=ddx(f)−ddx(g)

(4) Constant multiple rule: ddx(c⋅f)=c⋅ddx(f)

(5) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

**Calculation:**

Obtain the derivative of 2x2+x+xy=1.

Consider 2x2+x+xy=1.

Differentiate with respect to *x* on both sides,

ddx(2x2+x+xy)=ddx(1)

Apply the derivative rules (1),(2) and (4),

ddx(2x2)+ddx(x)+ddx(xy)=ddx(1)2ddx(x2)+ddx(x)+[xddx(y)+yddx(x)]=ddx(1)2(2x)+1+[xdydx+y(1)]=04x+1+y+xdydx=0

Simplify the terms and obtain dydx,

xdydx=−4x−1−ydydx=−4x−1−yxdydx=−(4x+1+y)x

Therefore, the differentiation of 2x2+x+xy=1 is dydx=−(4x+1+y)x.

#### To determine

**To find:** The equation explicitly for *y* and dydx.

#### Answer

The derivative of the equation 2x2+x+xy=1 is dydx=−x−2−2.

#### Explanation

**Given:**

The equation is 2x2+x+xy=1

**Calculation:**

The given equation can be expressed as follows,

xy=1−2x2−xy=1−2x2−xxy=1x−2x2x−1y=x−1−2x−1

The differentiate the equation y=x−1−2x−1 with respect to *x*,

dydx=ddx(x−1−2x−1)

Apply the derivative rule (3),(4) and (2),

dydx=ddx(x−1)−ddx(2x)−ddx(1)=ddx(x−1)−2ddx(x)−ddx(1)=(−1x−1−1)−2(1)−0=−x−2−2

Therefore, the derivative of the equation 2x2+x+xy=1 is dydx=−x−2−2.

#### To determine

**To check:** Whether the solutions from part (a) and part (b) are consistent or not.

#### Answer

The solutions from part (a) and part (b) are consistent.

#### Explanation

The derivative of the equation 2x2+x+xy=1 from part (a),

dydx=−(4x+1+y)x.

Substitute the value of y=x−1−2x−1(from part (b)),

dydx=−(4x+1+x−1−2x−1)x=−(4x+1+1x−2x−1)x=−(4x2+x+1−2x2−x)x2=−(1+2x2)x2

Simplify further,

dydx=−1−2x2x2=−1x2−2x2x2=−x−2−2

Since the derivative of the equation 2x2+x+xy=1 from part (a) is same as the derivative of the equation 2x2+x+xy=1 from part (b).

Therefore, it can be concluded that the two solutions from part (a) and (b) are consistent.