Use graphical differentiation to verify that StartFraction d e Superscript x Over dx EndFraction equals e Superscript x
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Part 1
Graph the function y equals e Superscript x. Choose the correct graph below. Each graph is shown on left bracket negative 1 comma 10 right bracket by left bracket negative 1 comma 10 right bracket.
A.

A coordinate system has a horizontal axis labeled from negative 1 to 10 in increments of 1 and a vertical axis labeled from negative 1 to 10 in increments of 1. A curve rises from left to right at an increasing rate passing through (0, 1) and approximately (2, 7.4).
Your answer is correct.B.

A coordinate system has a horizontal axis labeled from negative 1 to 10 in increments of 1 and a vertical axis labeled from negative 1 to 10 in increments of 1. A curve is above the positive horizontal axis and to the right of the positive vertical axis, approaching both.
C.

A coordinate system has a horizontal axis labeled from negative 1 to 10 in increments of 1 and a vertical axis labeled from negative 1 to 10 in increments of 1. A curve rises from left to right at a decreasing rate passing through (1, 0) and approximately (4, 1.4).
Part 2
Find the slope of the tangent line at x 1equalsminus5. Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice.
A.
The slope is

enter your response here.
​(Round to three decimal places as​ needed.)
B.
The slope is undefined.

### Part 1: Graph of the function \( y = e^x \)

The function \( y = e^x \) is an exponential function that grows rapidly as \( x \) increases. It has the following characteristics:
- It passes through the point \( (0, 1) \), because \( e^0 = 1 \).
- As \( x \) increases, the function grows faster and faster.
- As \( x \) decreases, the function approaches zero but never becomes negative.

Given these features, the correct graph is **A**, where the curve rises from left to right at an increasing rate and passes through the point \( (0, 1) \). 

### Part 2: Slope of the tangent line at \( x = -5 \)

The derivative of the function \( y = e^x \) is also \( e^x \). This means that the slope of the tangent line at any point \( x = a \) is given by \( e^a \).

For \( x = -5 \), the slope of the tangent line is:
\[
\text{Slope} = e^{-5}
\]
Using a calculator, we can compute \( e^{-5} \). Approximating to three decimal places:

\[
e^{-5} \approx 0.007
\]

So, the correct choice is **A**, with the slope approximately \( 0.007 \).

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Would you like more details, or do you have any questions on this? Here are five follow-up questions for deeper understanding:

1. What is the general shape of the graph of \( e^x \) for negative values of \( x \)?
2. How does the derivative of \( e^x \) compare to the derivative of other exponential functions?
3. Why does \( e^x \) have the same derivative as its original function?
4. How does the slope of the tangent line at different points on the graph of \( e^x \) change as \( x \) increases?
5. What is the significance of the point \( (0, 1) \) on the graph of \( e^x \)?

**Tip:** The function \( e^x \) is one of the few functions whose derivative is the same as the function itself, making it particularly important in calculus and exponential growth modeling.

Use graphical differentiation to verify that d/dx(e^x) equals e^x. Part 1: Graph the function y = e^x. Choose the correct graph. Part 2: Find the slope of the tangent line at x = -5.

Learn how to use graphical differentiation to verify that the derivative of e^x equals e^x. This includes graphing the exponential function and finding the slope of the tangent line at specific points like x = -5. Perfect for students in Grades 10-12 studying exponential functions and calculus.

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