Derivatives and Chain Rule Concepts

The video tutorial explains how to find the derivative of composite functions using the chain rule. It provides two examples: one with f(x) = sin(2ln(x)) and another with g(x) = cos(-3ln(x)). The process involves identifying the inner function, calculating its derivative, and applying the chain rule to find the derivative of the entire function. The tutorial concludes with a summary of the steps and encourages viewers to apply these techniques to similar problems.

What is the first step in finding the derivative of a composite function?

• Identify the outer function

• Identify the inner function

• Differentiate the outer function

• Differentiate the inner function

In the function f(x) = sin(2 ln(x)), what is the inner function?

• x

• 2 ln(x)

• cos(x)

• sin(x)

What is the derivative of 2 ln(x) with respect to x?

• ln(x)

• 2/x

• 1/x

• 2x

How do you express the derivative of f(x) = sin(2 ln(x)) using the chain rule?

• sin(2 ln(x)) * x

• sin(2 ln(x)) * 2/x

• cos(2 ln(x)) * 2/x

• cos(2 ln(x)) * x

What is the simplified form of f'(x) for f(x) = sin(2 ln(x))?

• 2 cos(2 ln(x))/x

• sin(2 ln(x))/x

• cos(2 ln(x))/x

• 2 sin(2 ln(x))/x

In the function g(x) = cos(-3 ln(x)), what is the inner function?

• sin(x)

• x

• -3 ln(x)

• cos(x)

What is the derivative of -3 ln(x) with respect to x?

• 3/x

• -3/x

• -ln(x)

• 3 ln(x)

How do you express the derivative of g(x) = cos(-3 ln(x)) using the chain rule?

• cos(-3 ln(x)) * 3/x

• -sin(-3 ln(x)) * -3/x

• -sin(-3 ln(x)) * 3/x

• cos(-3 ln(x)) * -3/x

What is the simplified form of g'(x) for g(x) = cos(-3 ln(x))?

• sin(-3 ln(x))/x

• -3 sin(-3 ln(x))/x

• 3 sin(-3 ln(x))/x

• -sin(-3 ln(x))/x

What is the result of multiplying two negative numbers in the context of g'(x)?

• Zero

• Undefined

• Negative

• Positive