Understanding Derivatives of Exponential and Logarithmic Functions

This lesson covers how to determine the derivatives of exponential and logarithmic functions with base e using the chain rule. It explains the process of finding the derivative of a composite function, specifically focusing on two examples: an exponential function and a logarithmic function. The lesson emphasizes the importance of identifying the inner function (U) and its derivative (U Prime) to apply the chain rule effectively. By the end of the lesson, students should understand how to apply the chain rule to differentiate these types of functions.

What is the main focus of this lesson?

• Solving quadratic equations

• Finding integrals of trigonometric functions

• Determining derivatives of exponential and logarithmic functions

• Graphing linear functions

Which rule is necessary to find the derivative of f(x) = 2 * e^(4x^2 + 2)?

• Power Rule

• Chain Rule

• Quotient Rule

• Product Rule

In the function f(x) = 2 * e^(4x^2 + 2), what is the expression for U?

• 2x^2 + 4

• x^2 + 4

• 2x + 4

• 4x^2 + 2

What is the derivative of U, U', for the function f(x) = 2 * e^(4x^2 + 2)?

• 16x

• 4x

• 8x

• 2x

What is the simplified form of the derivative of f(x) = 2 * e^(4x^2 + 2)?

• 4x * e^(4x^2 + 2)

• 2x * e^(4x^2 + 2)

• 8x * e^(4x^2 + 2)

• 16x * e^(4x^2 + 2)

What is the function g(x) in this lesson?

• -5 ln(3x + 5)

• 5 ln(3x + 5)

• 3 ln(5x + 3)

• -3 ln(5x + 3)

Which rule is used to find the derivative of g(x) = -3 ln(5x + 3)?

• Quotient Rule

• Product Rule

• Chain Rule

• Power Rule

What is the expression for U in the function g(x) = -3 ln(5x + 3)?

• 5x + 3

• 3x + 5

• x + 5

• 5x + 2

What is the derivative of U, U', for the function g(x) = -3 ln(5x + 3)?

• 4

• 2

• 3

• 5

What is the simplified form of the derivative of g(x) = -3 ln(5x + 3)?

• -10 / (5x + 3)

• -5 / (5x + 3)

• -3 / (5x + 3)

• -15 / (5x + 3)