Differentiation and Logarithmic Functions

The video tutorial explains how to find the derivative of a composite function, specifically f(x) = log base 3 of (2x e^x + 1). It begins by identifying the inner function and applying the chain rule to find its derivative, u prime. The product rule is then used to differentiate the expression 2x e^x. The tutorial continues by calculating u prime and applying the derivative formula for logarithms. Finally, the derivative is simplified to its final form, factoring out common terms. The video provides a step-by-step approach to understanding and applying these calculus concepts.

What is the base of the logarithm in the function f(x) = log base 3 of (2x e^x + 1)?

• e

• 3

• 10

• 2

Which rule is necessary to differentiate the inner function 2x e^x + 1?

• Power Rule

• Product Rule

• Chain Rule

• Quotient Rule

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

• x e^x

• e^x

• 0

• 1

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

• 1

• x

• 2

• 0

What is the derivative of a constant, such as 1, with respect to x?

• e^x

• x

• 0

• 1

What is the expression for u' when u = 2x e^x + 1?

• e^x + 1

• 2x e^x

• 2x e^x + 2 e^x

• 2x + e^x

What is the derivative formula for log base a of u with respect to x?

• u' * ln(a)

• 1/u * ln(a)

• ln(a) * u'

• 1/(ln(a) * u) * u'

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

• x+1

• 2e^x

• e^x

• 2x

In the final form of the derivative, what is the denominator?

• 2e^x

• ln(3) * (2x e^x + 1)

• 2x e^x + 1

• ln(3)

What is the final form of the derivative f'(x)?

• 2x e^x/(ln(3))

• e^x/(ln(3)(2x e^x + 1))

• 2e^x(x+1)/(ln(3)(2x e^x + 1))

• 2e^x/(ln(3))