https://file.aiquickdraw.com/mathbot/file/1729481693027nsxp5vcv.jpg

Let's carefully analyze the statements and find which is false.

### Statement (1):
\[
\frac{d}{dx} \left( a^x \right) = x a^{x-1} \quad \text{for all real} \quad a \geq 0.
\]
This rule is actually incorrect. The correct derivative of \( a^x \) is:
\[
\frac{d}{dx} \left( a^x \right) = a^x \ln(a),
\]
where \( a > 0 \), and \( a = 1 \) is a special case because \( 1^x = 1 \) for any \( x \), and its derivative is 0. Therefore, **statement (1) is false**.

### Statement (2):
Any line tangent to the graph of \( y = e^x \) intersects the graph at the point \( (\ln m, m) \), where \( m \) is the slope of the line.
To verify, we take the derivative of \( y = e^x \), which is \( \frac{dy}{dx} = e^x \). At any point \( (x_0, e^{x_0}) \), the slope of the tangent line is \( m = e^{x_0} \). The line will intersect the curve \( y = e^x \) at the point \( (\ln m, m) \) because if \( m = e^{x_0} \), we have \( x_0 = \ln m \). Therefore, **statement (2) is true**.

### Conclusion:
The false statement is **only (1)**. Therefore, the correct answer is:
\[
\boxed{\text{Only (1)}}
\]

Would you like more details or have any questions?

Here are five relative questions that expand on the concepts discussed:
1. What is the correct formula for the derivative of exponential functions like \( e^x \)?
2. How do we find the equation of a tangent line to a function at a given point?
3. Can the derivative formula for \( a^x \) apply to negative values of \( a \)?
4. How does the derivative of a natural logarithmic function \( \ln(x) \) behave?
5. What is the geometric interpretation of a tangent line on a curve?

**Tip**: For exponential functions \( a^x \), the logarithmic base \( \ln(a) \) always appears in the derivative formula, which is key to distinguishing their behavior from polynomials.

Which of the following statements is false? (1) The rule d/dx a^x = xa^{x-1} is true for all real a ≥ 0. (2) Any line tangent to the graph of y = e^x intersects the graph at the tangency point (ln m, m) where m is the slope of the line.

This calculus problem addresses the derivative rule for a^x and the tangent line to the exponential function y = e^x. Learn how to verify the correct derivative and the properties of the tangent line. Suitable for high school students in Grades 11-12.

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