Understanding Derivatives of Composite Functions

The video tutorial explains how to find the derivative of a composite function, specifically Y = log base 4 of (x^2 + x). It begins by identifying the composite nature of the function and defining U of X as x^2 + x. The derivative of U of X is found using the power rule. Next, V of X is defined as log base 4 of X, and its derivative is determined using the change of base formula. The chain rule is then applied to find the derivative of Y with respect to X, resulting in the expression (2x + 1) / (ln(4) * (x^2 + x)).

What is the initial expression for y in the problem?

• y = log base 2 of x + 2

• y = log base 4 of x + 2

• y = log base 2 of x^2 + x

• y = log base 4 of x^2 + x

What is U(x) defined as in the composite function?

• x^2 + 2

• x + 2

• x^2 - x

• x^2 + x

What is the derivative of U(x) using the power rule?

• x + 2

• 2x + 1

• x^2 + 1

• 2x - 1

How is V(x) defined in the context of this problem?

• log base 2 of x

• log base 4 of x

• log base 2 of x^2

• log base 4 of x^2

What is the derivative of V(x) in terms of natural logarithms?

• ln(2) / x

• ln(4) / x

• 1 / (x * ln(4))

• 1 / (x * ln(2))

Which rule is applied to find the derivative of y with respect to x?

• Chain rule

• Product rule

• Quotient rule

• Power rule

What expression represents the derivative of y with respect to x after applying the chain rule?

• (2x + 1) / (ln(4) * (x^2 + x))

• (2x - 1) / (ln(4) * (x^2 + x))

• (x^2 + x) / (ln(4) * (2x + 1))

• (x^2 - x) / (ln(4) * (2x + 1))

What is the role of the natural log of 4 in the derivative expression?

• It scales the derivative of the log function.

• It is added to the derivative.

• It is subtracted from the derivative.

• It is multiplied by the derivative.

What is the final form of the derivative of y with respect to x?

• (2x + 1) / (ln(4) * (x^2 + x))

• (2x - 1) / (ln(4) * (x^2 + x))

• (x^2 + x) / (ln(4) * (2x + 1))

• (x^2 - x) / (ln(4) * (2x + 1))

What mathematical concept is primarily used in this problem?

• Limits of composite functions

• Series expansion of composite functions

• Integration of composite functions

• Derivatives of composite functions