#### To determine

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

#### Answer

The implicit differentiation of 2x−1y=4 is dydx=2y2x2.

#### Explanation

**Given:**

The equation 2x−1y=4.

**Derivative rules:**

(1) Chain rule: If y=f(u) and u=g(x) are both differentiable function, then

dydx=dydu⋅dudx.

(2) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]−f1(x)ddx[f2(x)][f2x]2.

**Calculation:**

Obtain the derivative of the given equation.

2x−1y=4

Differentiate with respect to *x* on both sides,

ddx(2x−1y)=ddx(4)ddx(2x−1−y−1)=ddx(4)ddx(2x−1)−ddx(y−1)=ddx(4)2ddx(x−1)−ddx(y−1)=ddx(4)

Apply the chain rule (1) and simplify the terms,

2(−1x−1−1)−[ddy(y−1)dydx]=02(−1x−1−1)−[(−1y−1−1)dydx]=0−2x−2+y−2dydx=0y−2dydx=2x−2

Divided by y−2 on both sides,

dydx=2x−2y−2dydx=2y2x2

Therefore, the differentiation of 2x−1y=4 is dydx=2y2x2.

#### To determine

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

#### Answer

The derivative of the equation 2x−1y=4 is dydx=12(1−2x)2.

#### Explanation

**Given:**

The equation 2x−1y=4.

**Calculation:**

The given equation can be expressed explicitly for *y* as follows,

2x−1y=41y=2x−41y=2−4xxy=x2−4x

Differentiate on both sides with respect to *x*,

dydx=ddx(x2−4x)

Apply the Quotient rule (2) and simplify further,

dydx=[(2−4x)ddx(x)]−[(x)ddx(2−4x)](2−4x)2=[(2−4x)(1)]−[(x)[0−ddx(4x)]](2−4x)2=[(2−4x)(1)]−[(x)[0−4(1)]](2−4x)2=2−4x+4x(2−4x)2

Simplify further and obtain the value dydx,

dydx=2(2−4x)2=2(2(1−2x))2=24(1−2x)2=12(1−2x)2

Therefore, the derivative of the equation 2x−1y=4 is dydx=12(1−2x)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 2x−1y=4 from part (a),

dydx=2y2x2

Substitute the value y=x2−4x (from part (b)),

dydx=2(x2−4x)2x2=2x2x2(2−4x)2=24(1−2x)2=12(1−2x)2

Since the derivative of the equation 2x−1y=4 from part (a) is same as the derivative of the equation 2x−1y=4 from part (b).

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