Understanding Derivatives of Trigonometric Functions

The video tutorial explains how to find the derivative of a trigonometric function, f(x) = sin(x) * sin(x) * tan(x). Initially, the function is simplified using trigonometric identities, transforming it into sin^2(x). The chain rule is then applied to find the derivative, with the inner function being sin(x) and the outer function being the square function. The derivative is calculated as 2sin(x)cos(x), which can also be expressed using the double angle identity as sin(2x). The tutorial emphasizes the importance of simplification before applying differentiation rules.

What is the initial function given in the problem?

• tan x * sin x * cos x

• sin x * sin x * tan x

• cos x * tan x * sin x

• sin x * cos x * tan x

Which trigonometric identity is used to simplify the function?

• cos x = sin x / tan x

• tan x = cos x / sin x

• tan x = sin x / cos x

• sin x = 1/cos x

What is the simplified form of the function before differentiation?

• sin x * cos x

• cos^2 x

• sin^2 x

• tan^2 x

What is the inner function when applying the chain rule?

• tan x

• x^2

• sin x

• cos x

What is the derivative of the inner function sin x?

• sec x

• tan x

• cos x

• sin x

What rule is applied after identifying the inner and outer functions?

• Quotient Rule

• Product Rule

• Power Rule

• Chain Rule

What is the derivative of sin^2 x using the chain rule?

• sin^2 x

• 2 sin x cos x

• sin x cos x

• 2 cos x

Which identity is used to express the derivative in an alternative form?

• sin 2x = 2 sin x cos x

• sin 2x = sin^2 x + cos^2 x

• cos 2x = 2 cos^2 x - 1

• tan 2x = 2 tan x / (1 - tan^2 x)

What is the alternative expression for the derivative using the double angle identity?

• sin 2x

• sec 2x

• cos 2x

• tan 2x

Which of the following is a correct form of the derivative function?

• cos x - sin x

• tan x / sin x

• sin x + cos x

• 2 sin x cos x