Understanding Derivatives of Composite Functions

The video tutorial explains how to determine the derivative of a composite function, specifically f(x) = 3e^(sqrt(x)). It begins by identifying the function as composite and setting the inner function as u. The chain rule is applied to find the derivative, with u being the square root of x. The derivative of the inner function, u prime, is calculated as 1/(2sqrt(x)). Finally, the derivative of the original function is expressed as (3e^(sqrt(x)))/(2sqrt(x)). The tutorial emphasizes recognizing composite functions and applying the chain rule effectively.

What is the main goal of the problem discussed in the video?

• To solve a quadratic equation

• To calculate the area under a curve

• To determine the derivative of a composite function

• To find the integral of a function

What is the first step in dealing with a composite function?

• Identify the outer function

• Simplify the expression

• Find the integral of the function

• Identify the inner function

What substitution is made to simplify the expression?

• Let u = x^2

• Let u = ln(x)

• Let u = square root of x

• Let u = x^3

Which rule is applied to find the derivative of the composite function?

• Power Rule

• Product Rule

• Quotient Rule

• Chain Rule

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

• e^u * u'

• u' * e

• u * e^u

• e^u + u'

How is the derivative of the inner function, u, calculated?

• u' = 1/2 * x^(-1/2)

• u' = x^2

• u' = ln(x)

• u' = x^3

What is the simplified form of 1/2 * x^(-1/2)?

• 2 * x^(1/2)

• 1/(2 * x^(1/2))

• 1/2 * x^(1/2)

• x^(1/2) / 2

What is the final expression for the derivative of the function?

• 3e^(sqrt(x)) / (2 * sqrt(x))

• 3e^(x) / (2 * x^(1/2))

• 3e^(x^2) / (2 * x)

• 3e^(x^(1/2)) / (2 * x)

What is the key to solving the derivative problem discussed?

• Recognizing it as a composite function

• Using the product rule

• Finding the integral first

• Simplifying the expression

What is the overall difficulty level of the derivative problem once the composite function is recognized?

• Moderately difficult

• Very difficult

• Impossible to solve

• Fairly straightforward