#### To determine

**To find:** The derivative of the function.

#### Answer

The derivative of the function y=xg(x) is dydx=xg′(x)+g(x).

#### Explanation

**Given:**

The function y=xg(x).

**Derivative rule:**

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

(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

(3) Power rule: ddx(xn)=nxn−1

**Calculation:**

The derivative of the function y=xg(x) is dydx , which is obtained as follows,

dydx=ddx(xg(x))

Apply the product rule (1) and the power rule (3),

dydx=ddx(xg(x))=xddx(g(x))+g(x)ddx(x)=xg′(x)+g(x)(1)=xg′(x)+g(x)

Therefore, the derivative of the function y=xg(x) is dydx=xg′(x)+g(x).

#### To determine

**To find:** The derivative of the function.

#### Answer

The derivative of the function y=xg(x) is dydx=g(x)−xg′(x)[g(x)]2.

#### Explanation

**Given:**

The function y=xg(x).

**Calculation:**

The derivative of the function y=xg(x) is dydx , which is obtained as follows,

dydx=ddx(xg(x))

Apply the quotient rule (2) and the power rule (3),

dydx=g(x)ddx(x)−xddx(g(x))[g(x)]2=g(x)(1)−xg′(x)[g(x)]2=g(x)−xg′(x)[g(x)]2

Therefore, the derivative of the function y=xg(x) is dydx=g(x)−xg′(x)[g(x)]2.

#### To determine

**To find:** The derivative of the function.

#### Answer

The derivative of the function y=g(x)x is dydx=xg′(x)−g(x)x2.

#### Explanation

**Given:**

The function y=g(x)x

**Calculation:**

The derivative of the function y=g(x)x is dydx , which is obtained as follows,

dydx=ddx(g(x)x)

Apply the quotient rule (2) and the power rule (3),

dydx=xddx(g(x))−g(x)ddx(x)[x]2=xg′(x)−g(x)(1)x2=xg′(x)−g(x)x2

Therefore, the derivative of the function y=g(x)x is dydx=xg′(x)−g(x)x2.