#### To determine

**To sketch:** The graph of the function f(x)=x|x|.

#### Explanation

**Graph:**

Use the online graphing calculator to draw the graph of f(x)=x|x| as shown in Figure 1,

The graph of the equation is x2 above the y-axis while it is −x2 below the y-axis.

Thus, f(x)={x2if x≥0−x2if x<0

#### To determine

**To find:** The values of the function f(x)=x|x| is differentiable.

#### Answer

The function f(x)=x|x| is differentiable at all x∈ℝ.

#### Explanation

**Given:**

The function f(x)=x|x|.

**Result Used:**

The function f(x)=x2 is differentiable for all x∈ℝ.

**Calculation:**

Consider the function f(x)=x|x| or f(x)={x2if x≥0−x2if x<0.

By result, f(x)=x2 is differentiable for all x∈ℝ.

This implies that, f(x)=−x2 is also differentiable for all x∈ℝ.

Hence, f(x)=x|x| is differentiable at all points except 0.

Since the function |x| is not differentiable at x=0.

Check the differentiability of f(x)=x|x| at x=0.

f′(0)=limh→0f(0+h)−f(0)h=limh→0h|h|−0|0|h=limh→0|h|

Since |h| tends to 0 as *h* tends to 0, f′(0)=0.

Hence, f(x)=x|x| is differentiable at x=0.

Therefore, the function f(x)=x|x| is differentiable at all x∈ℝ.

#### To determine

**To find:** The formula for function f′(x).

#### Answer

The formula for function., f′(x)=2|x|

#### Explanation

**Given:**

The function is f(x)=x|x|.

**Calculation:**

Take derivative of the function f(x)=x|x| by considering the following cases,

**Case 1:** x≥0

d{f(x)}dx=ddx(x|x|) =limh→0(x+h)|x+h|−x|x|h

Since x≥0, then x+h≥0 implying |x+h|=x+h and |x|=x.

d{f(x)}dx=limh→0(x+h)2−x2h=limh→0h2+2xhh=limh→0(h+2x)=2x

Thus, it can be conclude that f′(x)=2x if x≥0.

**Case 2:** x<0

d{f(x)}dx=ddx(x|x|)=limh→0(x+h)|x+h|−x|x|h

Since x<0, x+h also less than zero.

This implies that, |x+h|=−(x+h) and |x|=−x.

d{f(x)}dx=limh→0−(x+h)2+x2h =limh→0−h2−2xhh =limh→0(−h−2x) =−2x

Thus, it can be conclude that f′(x)=2x if x<0.

Therefore, the formula of derivative is f′(x)={2xif x≥0−2xif x<0

But the derivative function can be rewritten as

f′(x)={2xif x≥0−2xif x<0=2{xif x≥0−xif x<0=2|x|

Thus, the formula for the function f′(x)=2|x|.