https://file.aiquickdraw.com/mathbot/file/1725512163019udbtj2wn.jpg

In the image, the graph shows the function \( f(x) = 10^x \), and a tangent line is drawn to the curve. The task is to find the slope \( m \) of this tangent line.

To find the slope of the tangent line at a specific point on the curve of \( f(x) = 10^x \), we first need to compute the derivative of the function \( f(x) \), as the derivative gives us the slope of the tangent line at any point \( x \).

The derivative of \( f(x) = 10^x \) is:

\[
f'(x) = \frac{d}{dx} \left( 10^x \right) = 10^x \ln(10)
\]

From the graph, it looks like the tangent line is drawn at \( x = 1 \). Therefore, we substitute \( x = 1 \) into the derivative:

\[
f'(1) = 10^1 \ln(10) = 10 \ln(10)
\]

Now we calculate the numerical value of this expression:

\[
f'(1) = 10 \times 2.3026 \approx 23.03
\]

Thus, the slope \( m \) of the tangent line at \( x = 1 \) is approximately \( 23.03 \).

Would you like more details on the steps or any further explanation?

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Here are five related questions:
1. How would the slope change if the tangent was drawn at \( x = 2 \)?
2. What is the significance of the logarithmic constant \( \ln(10) \) in this context?
3. How do you interpret the meaning of a tangent line on a curve graphically?
4. Can you find the equation of the tangent line given the slope and a point on the curve?
5. What is the derivative of other exponential functions like \( 2^x \)?

**Tip:** When calculating derivatives of exponential functions, remember to multiply by the natural logarithm of the base.

Find the equation of the line in the figure below. Give exact values for the form y = mx + b.

Learn how to find the slope of the tangent line for the exponential function f(x) = 10^x at x = 1. This detailed solution covers the derivative of the function and calculating the exact slope of the tangent. Suitable for Grades 11-12 calculus students.

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