Derivatives of Exponential Functions

What rule is applied to find the derivative of a product of two functions?

• Product Rule

• Chain Rule

• Quotient Rule

• Power Rule

Which of the following is NOT a step in applying the product rule?

• Finding the derivative of the first function

• Adding the products of derivatives

• Finding the derivative of the second function

• Multiplying the derivatives together

What is the derivative of an exponential function a^u with respect to x?

• a^u * du/dx

• ln(a) * a^u

• a^u * u

• a^u * ln(a) * du/dx

What is the derivative of the exponent 4x^2 in the function 5^(4x^2)?

• 8x

• 4x

• 2x

• 16x

In the context of the video, what is the derivative of 5^(4x^2)?

• ln(5) * 5^(4x^2)

• 5^(4x^2) * 4x

• 8x * 5^(4x^2)

• 5^(4x^2) * ln(5) * 8x

What is the role of natural logarithms in finding the derivative of exponential functions?

• They are used to multiply with the original function

• They are used to find the derivative of the exponent

• They are used to simplify the expression

• They are used to change the base of the exponentials

What is the derivative of 2^(3x) as explained in the video?

• 2^(3x) * 3

• ln(2) * 2^(3x)

• 3 * 2^(3x)

• 2^(3x) * ln(2) * 3

How is the product rule applied to the function 2^(3x) * 5^(4x^2)?

• By applying the chain rule to each function

• By multiplying the derivatives of both functions

• By adding the derivatives of both functions

• By using the product rule to combine derivatives

What common factor is factored out in the final expression of the derivative?

• 3x and 4x^2

• 2^(3x) and 5^(4x^2)

• 8x and 3

• ln(5) and ln(2)

What is the purpose of factoring out common terms in the derivative expression?

• To eliminate the logarithms

• To make the expression longer

• To simplify the expression

• To change the base of the exponentials