Differentiating Exponential Functions

The video tutorial covers the concept of natural logarithms and the number e, explaining how e is related to exponential growth. It delves into the derivative of e^x, showing that it remains unchanged. The tutorial also discusses the gradient and tangent line of e^x, emphasizing that the gradient at any point is equal to its y-value. The chain rule is applied to differentiate e^2x, illustrating the process with substitutions. Finally, an example of differentiating e^(x^2) is provided, demonstrating the differentiation of both the inside and outside functions.

What is the base of the natural logarithm?

• π

• 10

• e

• 2

What is the derivative of e^x?

• e^2

• x^2

• x^e

• e^x

At x = 0, what is the y-intercept of e^x?

• x

• e

• 1

• 0

What is the gradient of e^x at any point?

• Zero

• One

• Its y-value

• Its x-value

When applying the chain rule to e^(2x), what is the derivative?

• 2e^(2x)

• e^(2x)

• 2x

• e^x

In the chain rule, what does the 'inside function' refer to?

• The constant

• The derivative

• The exponent

• The base

What substitution is made when using the chain rule for e^(2x)?

• u = e^x

• u = x^2

• u = 2

• u = 2x

How do you differentiate e^(x^2) using the chain rule?

• 2xe^(x^2)

• e^(x^2)

• x^2e^x

• 2e^x

What is the derivative of the inside function x^2?

• 2x

• x

• x^2

• 2

What remains unchanged when differentiating e to the power of something?

• The constant

• The base

• The exponent

• The derivative