Trig Chain Rule Mastery

Find the derivative of y = sin(3x) using the chain rule.

• sin(3x)

• 3cos(x)

• 3cos(3)

• 3cos(3x)

Answer explanation

The derivative of sin(3x) using the chain rule is 3cos(3x), where the derivative of sin(u) is cos(u) multiplied by the derivative of the inside function, which is 3.

Calculate the derivative of y = cos(2x) using the chain rule.

• cos(2x)

• -2cos(2x)

• 2sin(2x)

• -2sin(2x)

Answer explanation

The derivative of cos(2x) using the chain rule is -2sin(2x), so the correct answer is -2sin(2x).

Determine the derivative of y = tan(4x) using the chain rule.

• y' = 4sin(4x)

• y' = 4tan^2(4x)

• y' = 4sec^2(4x)

• y' = 4cos(4x)

Answer explanation

The derivative of y = tan(4x) using the chain rule is y' = 4sec^2(4x). This is because the derivative of tan(u) is sec^2(u), and the derivative of 4x is 4.

Find the derivative of y = sin(2x) * cos(3x) using the product rule.

• The derivative of y = sin(2x) * cos(3x) is y' = (2cos(2x)cos(3x)) - (3sin(2x)sin(3x))

• y' = (2cos(2x)sin(3x)) - (3sin(2x)cos(3x))

• y' = (2sin(2x)cos(3x)) - (3cos(2x)sin(3x))

• y' = (2sin(2x)sin(3x)) - (3cos(2x)cos(3x))

Answer explanation

The correct derivative using the product rule is y' = (2cos(2x)cos(3x)) - (3sin(2x)sin(3x))

Calculate the derivative of y = cos(2x) / sin(3x) using the quotient rule.

• y' = (2sin(2x)sin(3x) - cos(2x)3sin(3x)) / sin^2(3x)

• y' = (2sin(2x)sin(3x) + cos(2x)3cos(3x)) / sin^2(3x)

• y' = (2cos(2x)sin(3x) - cos(2x)3cos(3x)) / sin^2(3x)

• y' = (2sin(2x)sin(3x) - cos(2x)3cos(3x)) / sin^2(3x)

Answer explanation

The correct choice is y' = (2sin(2x)sin(3x) - cos(2x)3cos(3x)) / sin^2(3x) because it is the result of applying the quotient rule to the given function.

Determine the derivative of y = tan(4x) * cos(3x) using the product rule.

• 4sec^2(4x)cos(3x) - 3cot(4x)sin(3x)

• 4sec^2(4x)cos(3x) - 3tan(4x)sin(3x)

• 4sec^2(4x)cos(3x) + 3tan(4x)sin(3x)

• 4sec^2(4x)sin(3x) - 3tan(4x)cos(3x)

Answer explanation

The derivative of y = tan(4x) * cos(3x) using the product rule is 4sec^2(4x)cos(3x) - 3tan(4x)sin(3x), which matches the first answer choice.

Find the derivative of y = sin(2x) / cos(3x) using the quotient rule.

• (2cos(2x)cos(3x) - sin(2x)(-3sin(3x))) / (cos(3x))^2

• 2cos(2x)cos(3x) + sin(2x)(-3sin(3x))

• cos(2x)cos(3x) - sin(2x)(-3sin(3x))

• 2cos(2x)cos(3x) - sin(2x)(-3sin(3x))

Answer explanation

The correct answer is (2cos(2x)cos(3x) - sin(2x)(-3sin(3x))) / (cos(3x))^2, which is obtained using the quotient rule for differentiation.

Calculate the derivative of y = tan(4x) / sin(3x) using the quotient rule.

• 4sec^2(4x)sin(3x) - tan(4x)3cos(3x) / sin^2(3x)

• 4sec^2(4x)sin(3x) - tan(4x)3cos(3x) / cos^2(3x)

• 4sec^2(4x)sin(3x) - tan(4x)3cos(3x) / sin(3x)

• (4sec^2(4x)sin(3x) - tan(4x)3cos(3x)) / sin^2(3x)

Answer explanation

The correct answer is (4sec^2(4x)sin(3x) - tan(4x)3cos(3x)) / sin^2(3x) because it follows the quotient rule for finding the derivative of the given function.

Determine the derivative of y = sin(cos(2x)) using the chain rule.

• y' = -2cos(2x)sin(cos(x))

• y' = -2cos(cos(2x))cos(2x)

• y' = -2cos(cos(2x))sin(2x)

• y' = -2sin(cos(2x))cos(2x)

Answer explanation

The correct choice is y' = -2cos(cos(2x))sin(2x) because when using the chain rule, the derivative of sin(u) is cos(u) * u' and the derivative of cos(2x) is -2sin(2x).

Find the derivative of y = cos(tan(3x)) using the chain rule.

• -3cos(tan(3x))sec^2(3x)

• -3sin(tan(3x))csc^2(3x)

• -3sin(tan(3x))sec^2(3x)

• 3sin(tan(3x))sec^2(3x)

Answer explanation

The derivative of y = cos(tan(3x)) using the chain rule is -3sin(tan(3x))sec^2(3x), which is the correct answer choice.