Understanding Derivatives of Exponential Functions

This video tutorial covers the differentiation of exponential functions, focusing on the derivative of e^x and other bases. It includes a proof of why the derivative of e^x is e^x, explores derivatives with non-e bases, and provides practical examples using product and quotient rules. The video concludes with real-world applications, such as calculating the rate of change in spending on organic food.

What is the derivative of e^x?

• 1/x

• x^e

• e^x

• ln(x)

When applying the chain rule to e^(u(x)), what is the derivative?

• e^(u(x)) * u'(x)

• u(x) * e^(u(x))

• e^(u(x)) + u'(x)

• u'(x) / e^(u(x))

If the base of an exponential function is not e, what additional factor appears in the derivative?

• Square of the base

• Natural log of the base

• Exponential of the base

• Reciprocal of the base

What is the special limit that defines the number e?

• The limit of e^n as n approaches zero

• The limit of n/e as n approaches infinity

• The limit of n^n as n approaches zero

• The limit of (1 + 1/n)^n as n approaches infinity

In the product rule, what is the derivative of a product of two functions f(x) and g(x)?

• f(x) + g(x)

• f'(x)g'(x)

• f(x)g(x)

• f'(x)g(x) + f(x)g'(x)

What is the derivative of e^(-2x) using the chain rule?

• -2e^(-2x)

• 2e^(-2x)

• e^(-2x)

• -e^(-2x)

What is the derivative of a quotient of two functions f(x) and g(x)?

• (g(x)f'(x) - f(x)g'(x)) / (g(x))^2

• (f(x)g(x)) / (g(x))^2

• (f(x)g'(x) + g(x)f'(x)) / (g(x))^2

• (f'(x)g'(x)) / (g(x))^2

How do you determine the slope of a tangent line to a function at a given point?

• Find the derivative and evaluate it at the point

• Find the integral and evaluate it at the point

• Multiply the function value by the point

• Divide the function value by the point

In the context of the video, what does the term 'rate of change' refer to?

• Derivative

• Integral

• Limit

• Product

How can a graphing calculator be used to verify the derivative at a specific point?

• By evaluating the derivative function at that point

• By plotting the function and checking its slope

• By integrating the function at that point

• By finding the limit of the function at that point