Understanding Derivatives of Composite Functions

The video tutorial explains how to find the derivatives of composite functions Big F of X and Big G of X using the chain rule. It begins by introducing the concept of composite functions and the chain rule, then applies these principles to derive Big F of X, where the inner function is X cubed. The tutorial continues with deriving Big G of X, where the inner function is f of X. The process involves identifying the inner and outer functions, applying the chain rule, and using given values to calculate specific derivatives.

What is the main mathematical concept used to find the derivatives of composite functions?

• Power Rule

• Chain Rule

• Quotient Rule

• Product Rule

In the context of Big F(x) = F(x^3), what is identified as the inner function?

• x

• F(x^3)

• x^3

• F(x)

How is the derivative of Big F(x) expressed using the chain rule?

• 3x^2 * F'(x^3)

• F'(x^3) * x^2

• 3 * F(x^3)

• F(x^3) * 3x

What is the value of Big F'(a) given a^2 = 12 and F'(a^3) = 4?

• 48

• 144

• 72

• 36

In the context of Big G(x) = (F(x))^3, what is identified as the outer function?

• F(x)

• x^3

• (F(x))^3

• x

Which rule is used to differentiate a function raised to a power in the context of the chain rule?

• Product Rule

• Quotient Rule

• Power Rule

• Sum Rule

How is the derivative of Big G(x) expressed using the chain rule?

• 3 * (F(x))^2 * F'(x)

• 3 * F(x) * x

• F(x) * F'(x)

• F'(x) * 3

What is the value of Big G'(a) given F(a) = 3 and F'(a) = 2?

• 36

• 72

• 54

• 18

What is the derivative of the inner function U when U = F(x) in the context of Big G(x)?

• F'(x)

• F(x)

• x

• 3x^2

What is the role of the derivative of the inner function in the chain rule?

• It is multiplied by the outer function's derivative.

• It is divided by the outer function's derivative.

• It is added to the outer function's derivative.

• It is subtracted from the outer function's derivative.