Derivatives of Exponential Functions

The video tutorial explains how to find the derivative of a function using the product rule. It begins by introducing the concept of derivatives and the product rule, then applies the rule to a function involving exponential terms. The tutorial demonstrates how to find the derivative of each component and simplify the resulting expression. Finally, it shows how to factor out common terms to express the derivative in a simplified form.

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