Trigonometric Derivatives and Functions

The video tutorial explains how to find the derivative of a given function using simplification techniques and the product rule. It starts by simplifying the function using trigonometric identities, then applies the product rule to find the derivative. The derivative is evaluated at a specific point, and trigonometric values are calculated using the unit circle. Finally, the exact value and decimal approximation of the slope of the tangent line are determined.

What is the initial function given in the problem?

• F(x) = 3x^2 * sin(x) / cos(x)

• F(x) = 3x^2 * cos(x) / sin(x)

• F(x) = 3x^2 * sec(x) / tan(x)

• F(x) = 3x^2 * tan(x) / sec(x)

How is tangent expressed in terms of sine and cosine?

• tan(x) = cos(x) / sin(x)

• tan(x) = sin(x) / cos(x)

• tan(x) = sin(x) * cos(x)

• tan(x) = 1 / cos(x)

What is the simplified form of the function F(x) after removing the quotient?

• F(x) = 3x^2 * sec(x)

• F(x) = 3x^2 * tan(x)

• F(x) = 3x^2 * sin(x)

• F(x) = 3x^2 * cos(x)

Which rule is used to find the derivative of the simplified function?

• Quotient Rule

• Power Rule

• Product Rule

• Chain Rule

What is the derivative of sin(x)?

• cos(x)

• -sin(x)

• -cos(x)

• tan(x)

What is the derivative of 3x^2?

• 6x

• x^3

• 3x

• 9x^2

At which angle is the derivative evaluated to find the slope of the tangent line?

• π/4

• π/2

• π/6

• π/3

What is the cosine value at π/4 radians?

• 1

• 1/2

• √3/2

• √2/2

What is the exact value of the derivative at x = π/4?

• 3√2π/8

• 3π/4

• 3π/8

• 3√2/4

What is the approximate slope of the tangent line at x = π/4?

• 2.718

• 1.414

• 4.641

• 3.141