Question

Which of the following functions is differentiable on the set of real numbers?

• A. \(f(x)=[x]\)
• B. \(f(x)=|x|+|x+1|\)
• C. \(f(x)=\sin x+\log e^x+e^x\)
• D. \(f(x)=\dfrac{x}{|x|}\)

Hint:

Common non-differentiable functions: \[ |x|,\qquad [x],\qquad \frac{x}{|x|} \] Common everywhere differentiable functions: \[ e^x,\qquad \sin x,\qquad \cos x,\qquad \ln(e^x)=x \]

Answer 1

The Correct Option is C

Concept: A function is differentiable on \(\mathbb{R}\) if its derivative exists at every real number. Differentiability implies continuity.

Step 1: Check option (A). \[ f(x)=[x] \] The greatest integer function is discontinuous at every integer. Hence it is not differentiable on \(\mathbb R\).

Step 2: Check option (B). \[ f(x)=|x|+|x+1| \] The function contains cusps at \[ x=0 \] and \[ x=-1 \] Hence it is not differentiable everywhere.

Step 3: Check option (C). \[ f(x)=\sin x+\log e^x+e^x \] Since \[ \log e^x=x \] the function becomes \[ f(x)=\sin x+x+e^x \] Each term is differentiable for all real \(x\). Therefore, \[ f(x) \] is differentiable on \(\mathbb R\).

Step 4: Check option (D). \[ f(x)=\frac{x}{|x|} = \begin{cases} 1, & x\gt 0 \\ -1, & x\lt 0 \end{cases} \] It is not even defined at \[ x=0 \] Hence it cannot be differentiable on \(\mathbb R\).

Step 5: Select the correct option. \[ \begin{aligned} (A) &\rightarrow \text{Not differentiable on } \mathbb{R} \\ (B) &\rightarrow \text{Not differentiable on } \mathbb{R} \\ (C) &\rightarrow \text{Differentiable on } \mathbb{R} \\ (D) &\rightarrow \text{Not differentiable on } \mathbb{R} \end{aligned} \] \[\begin{aligned} \boxed{ f(x)=\sin x+\log e^x+e^x } \end{aligned}\] Hence, option \(\mathbf{(C)}\) is correct.