Understanding Derivatives and Chain Rule

The video tutorial explains how to find the first and second derivatives of the function f(x) = tan(2x) using the chain rule. It begins by identifying the function as a composite function and applies the chain rule to find the first derivative. The tutorial then evaluates the first derivative at x = π/8, showing that the slope of the tangent line is 4, indicating the function is increasing. Next, it finds the second derivative by applying the chain rule twice, demonstrating that the function is concave up at x = π/8. Finally, the tutorial verifies these results graphically.

What is the outer function in the given composite function?

• Cosine

• 2x

• Tangent

• Secant

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

• Chain Rule

• Product Rule

• Quotient Rule

• Power Rule

What is the value of the first derivative at x = π/8?

• 8

• 4

• 2

• 16

What does a positive first derivative indicate about the function at a point?

• The function is concave down

• The function is increasing

• The function is constant

• The function is decreasing

What is the inner function when finding the second derivative?

• Cosine 2x

• Secant 2x

• Tangent 2x

• 2x

How many times is the chain rule applied to find the second derivative?

• Three times

• Twice

• Once

• Not at all

What is the value of the second derivative at x = π/8?

• 12

• 16

• 4

• 8

What does a positive second derivative indicate about the function's concavity?

• Concave up

• Concave down

• Linear

• No concavity

What is the slope of the tangent line at x = π/8?

• 2

• 4

• 6

• 8

What does the graphical verification confirm about the function at x = π/8?

• The function is constant

• The function is increasing and concave up

• The function is decreasing

• The function is concave down