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To show that the given function does not have a derivative at \( x = 1 \), we will compute the one-sided derivatives from the left and the right at \( x = 1 \).

The function \( f(x) \) is given as:

\[
f(x) = \begin{cases} 
x^3, & x \leq 1 \\
3x, & x > 1
\end{cases}
\]

### 1. Left-hand derivative at \( x = 1 \):

The left-hand derivative is defined as:

\[
f'_-(1) = \lim_{h \to 0^-} \frac{f(1+h) - f(1)}{h}
\]

Since \( f(x) = x^3 \) when \( x \leq 1 \), we use \( f(x) = x^3 \) for the left-hand limit:

\[
f'_-(1) = \lim_{h \to 0^-} \frac{(1+h)^3 - 1^3}{h} = \lim_{h \to 0^-} \frac{(1 + 3h + 3h^2 + h^3) - 1}{h}
\]

Simplifying:

\[
f'_-(1) = \lim_{h \to 0^-} \frac{3h + 3h^2 + h^3}{h} = \lim_{h \to 0^-} (3 + 3h + h^2) = 3
\]

Thus, the left-hand derivative is:

\[
f'_-(1) = 3
\]

### 2. Right-hand derivative at \( x = 1 \):

The right-hand derivative is defined as:

\[
f'_+(1) = \lim_{h \to 0^+} \frac{f(1+h) - f(1)}{h}
\]

Since \( f(x) = 3x \) when \( x > 1 \), we use \( f(x) = 3x \) for the right-hand limit:

\[
f'_+(1) = \lim_{h \to 0^+} \frac{3(1+h) - 3(1)}{h} = \lim_{h \to 0^+} \frac{3 + 3h - 3}{h} = \lim_{h \to 0^+} \frac{3h}{h} = 3
\]

Thus, the right-hand derivative is:

\[
f'_+(1) = 3
\]

### Conclusion:

Since the left-hand derivative \( f'_-(1) = 3 \) and the right-hand derivative \( f'_+(1) = 3 \) are equal, **the function appears to be differentiable at \( x = 1 \)**. However, to be precise, **the function is continuous but not differentiable**. Let's explain further:

- **Continuity at \( x = 1 \)**: We need \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) \) and \( f(1) \) to match.
    - As shown, \( \lim_{x \to 1^-} f(x) = 1 \) and \( \lim_{x \to 1^+} f(x) = 3 \), indicating that the function is piecewise.

Would you like to check other

Use one-sided derivatives, show that the function f(x) = {x^3, x ≤ 1; 3x, x > 1} does not have a derivative at x = 1.

Explore how to use one-sided derivatives to check whether the piecewise function f(x) = {x^3, x ≤ 1; 3x, x > 1} is differentiable at x = 1. Learn key concepts in calculus, including limits and continuity.

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