Differentiation and Product Rule Concepts

The video tutorial explains how to differentiate a function using the product rule. It begins by introducing the function f(x) and the need to find its derivative, f'(x). The product rule is explained, which involves identifying two functions, U and V, and applying the rule to find the derivative. The tutorial then demonstrates the process of finding the derivatives of the individual functions and combining them according to the product rule. Finally, the derivative expression is simplified, and the tutorial concludes with the final form of the derivative.

What is the function f(x) that needs to be differentiated?

• 5 sin(x) + 3 cos(x) * tan^(-1)(x)

• 5 sin(x) - 3 cos(x) * tan^(-1)(x)

• 5 cos(x) - 3 sin(x) * tan^(-1)(x)

• 5 sin(x) * 3 cos(x) - tan^(-1)(x)

Which rule is used to differentiate the product of two functions?

• Quotient Rule

• Power Rule

• Chain Rule

• Product Rule

In the product rule, what is the expression for the derivative of U * V?

• U * V' + V * U'

• U * V + U' * V'

• U' * V + U * V'

• U' * V' + U * V

What is identified as the first function U in the product rule?

• 5 sin(x) + 3 cos(x)

• 5 sin(x) - 3 cos(x)

• 3 cos(x) - 5 sin(x)

• 3 sin(x) + 5 cos(x)

What is the derivative of inverse tangent x?

• 1 / (1 - x^2)

• 1 / (1 + x^2)

• x / (1 - x^2)

• x / (1 + x^2)

What is the derivative of 5 sin(x)?

• 5 cos(x)

• -5 sin(x)

• 5 sin(x)

• -5 cos(x)

What is the derivative of -3 cos(x)?

• -3 cos(x)

• 3 cos(x)

• -3 sin(x)

• 3 sin(x)

How is the first product in the derivative expression simplified?

• As a single fraction

• As a sum of fractions

• As a difference of fractions

• As a product of fractions

What is the final form of the derivative function?

• 5 sin(x) + 3 cos(x) / (1 + x^2) + tan^(-1)(x) * (5 sin(x) + 3 cos(x))

• 5 sin(x) - 3 cos(x) / (1 + x^2) + tan^(-1)(x) * (5 cos(x) + 3 sin(x))

• 5 sin(x) - 3 cos(x) / (1 + x^2) - tan^(-1)(x) * (5 cos(x) + 3 sin(x))

• 5 sin(x) + 3 cos(x) / (1 + x^2) - tan^(-1)(x) * (5 sin(x) + 3 cos(x))

What is the purpose of the tutorial?

• To simplify a trigonometric expression

• To find the derivative of a function using the product rule

• To solve a differential equation

• To find the integral of a function