Calculating Derivatives of Exponential Functions

The video tutorial explains how to find the first and second derivatives of a function using the product rule. It begins by identifying the function as a product of two functions and applies the product rule to find the first derivative. The tutorial then proceeds to calculate the second derivative by differentiating the first derivative. Throughout the process, the video emphasizes the importance of simplifying expressions and identifying common factors. The tutorial concludes with a simplified expression for the second derivative.

What is the initial function given in the problem?

• f(x) = e^x * sin(x)

• f(x) = 2e^x * cos(x)

• f(x) = 2e^x + cos(x)

• f(x) = 2e^x - cos(x)

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

• Chain Rule

• Quotient Rule

• Power Rule

• Product Rule

What is the derivative of cos(x) with respect to x?

• -cos(x)

• cos(x)

• -sin(x)

• sin(x)

What is the greatest common factor in the first derivative expression?

• e^x

• 2e^x

• cos(x)

• sin(x)

What is the expression for the first derivative after factoring?

• e^x * (cos(x) - sin(x))

• 2e^x * (cos(x) - sin(x))

• 2e^x * (cos(x) + sin(x))

• 2e^x * (sin(x) - cos(x))

What is the derivative of 2e^x with respect to x?

• e^x

• 2x

• 2

• 2e^x

What is the expression for the second derivative before simplification?

• 2e^x * (-sin(x) + cos(x))

• 2e^x * (sin(x) + cos(x))

• 2e^x * (sin(x) - cos(x))

• 2e^x * (-sin(x) - cos(x))

Which terms cancel out during the simplification of the second derivative?

• 2e^x sin(x) and -2e^x sin(x)

• -2e^x sin(x) and -2e^x cos(x)

• 2e^x cos(x) and -2e^x cos(x)

• 2e^x sin(x) and 2e^x cos(x)

What is the final expression for the second derivative?

• -2e^x sin(x)

• 2e^x sin(x)

• 4e^x sin(x)

• -4e^x sin(x)

What is the purpose of applying the product rule twice in this problem?

• To find the limit of the function

• To simplify the function

• To find the first and second derivatives

• To find the integral of the function