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Understanding Derivatives of Exponential and Logarithmic Functions in Brief Calculus

This section explores the derivatives of exponential functions, such as (f(x) = 2^x) and (f(x) = a^x), providing insights into their rate of change. Through graphical representations, we discuss the shape of the derivative graph (f'(x)) and calculate these derivatives to establish their relationships. The limit and value of the derivative, approximately 0.693 for (f(x) = 2^x), suggest a proportionality concept. Additionally, we numerically estimate the derivative of the natural logarithm function, (y = ln(x)), reinforcing our understanding of these fundamental concepts in calculus.

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Understanding Derivatives of Exponential and Logarithmic Functions in Brief Calculus

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  1. Section 3.3 Exponential and Logarithmic Rate-of-Change Formulas MAT 213 Brief Calculus

  2. Below is a graph of f(x)=2x What do you think the graph of f’(x) would look like?

  3. What is this???? The Derivatives of Exponential Functions Calculate the derivative of f(x)=2x

  4. The Derivatives of Exponential Functions Fill out the following table for values of h close to zero.

  5. The Derivatives of Exponential Functions Fill out the following table for values of h close to zero. This table suggests that the limit DOES exist, and has a value of about 0.693 So we can write:

  6. So the derivative of 2x is proportional to 2xwith a constant of proportionality 0.693. Hmmmm…

  7. The Derivatives of Exponential Functions Calculate the derivative of f(x)=ax What is this????

  8. Use your calculator to plot these points. What type of function does it look like? Here is for different values of a

  9. RESULTS Consequently,

  10. EXAMPLES

  11. The Derivative of ln x Numerically estimate the derivative at the following input values.

  12. The Derivative of ln x Numerically estimate the derivative at the following input values.

  13. The Derivative of ln x If y = lnx, then for x > 0. Examples

  14. In groups let’s try the following from the book • 23

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