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Let's analyze each of the options to determine which formula is **not** correct.

### A. \( \frac{d}{dx} c x = x c^{x-1} \)

This is incorrect. The derivative of \( c x \) with respect to \( x \), where \( c \) is a constant, should be \( 0 \) because the constant term does not depend on \( x \). The rule used here is incorrect.

### B. \( \frac{d}{dx} e^{cx} = c e^{cx} \)

This is correct. The derivative of an exponential function \( e^{cx} \) with respect to \( x \) is \( c e^{cx} \), following the chain rule.

### C. \( \frac{d}{dx} x^c = c x^{c-1} \)

This is correct. This is a standard power rule derivative for constants: \( \frac{d}{dx} x^c = c x^{c-1} \).

### D. \( \frac{d}{dx} \left( c \cdot \ln(x) \right) = \frac{c}{x} \)

This is correct. The derivative of \( \ln(x) \) is \( \frac{1}{x} \), so applying the constant multiple rule gives \( \frac{d}{dx} \left( c \cdot \ln(x) \right) = c \cdot \frac{1}{x} = \frac{c}{x} \).

### E. "None of the above. That is, all of the above are correct."

This is **false** because option A is incorrect.

### Conclusion: 
The incorrect formula is **A**.

Would you like further details on any of these steps?

Here are some related questions to expand on this topic:
1. How do you apply the chain rule to more complex exponential functions?
2. Why is the derivative of a constant term always zero?
3. Can the power rule be used for negative and fractional exponents as well?
4. How does the product rule apply if \( c \) were a function rather than a constant?
5. What are the conditions for using logarithmic differentiation in more advanced problems?

**Tip**: When differentiating functions with constants, always verify whether the constant affects the variable directly or indirectly to apply the correct differentiation rule.

If c is a constant, which of the following formulas is not correct?

Analyze the differentiation of constant terms and exponential functions with a step-by-step explanation. This problem covers key concepts such as the power rule, chain rule, and logarithmic differentiation, ideal for students in Grades 10-12.

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