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.