# Differentiation Rules of Sinusoidal Functions

• chudzoik
In summary, the correct answer for f'(x) when given f(x) = cos2x - sin2x is -2sin2x. This can be found by using the identity 2sin(x)cos(x) = sin(2x).
chudzoik

## Homework Statement

f(x) = cos2x - sin2x

## The Attempt at a Solution

f'(x) = (2cosx)(-sinx) - (2sinx)(cosx)
f'(x) = -2cosxsinx - 2sinxcosx

This is what I think the answer should be, but the back of the book says otherwise. I need help identifying what I did wrong.

chudzoik said:

## Homework Statement

f(x) = cos2x - sin2x

## The Attempt at a Solution

f'(x) = (2cosx)(-sinx) - (2sinx)(cosx)
f'(x) = -2cosxsinx - 2sinxcosx

This is what I think the answer should be, but the back of the book says otherwise. I need help identifying what I did wrong.
... And what would that otherwise be?

Perhaps something involving sine and/or cosine of 2x ?

... or just a simplified version of what you have?

-2ab - 2ba = -4ab ?

SammyS said:
... And what would that otherwise be?

Perhaps something involving sine and/or cosine of 2x ?

... or just a simplified version of what you have?

-2ab - 2ba = -4ab ?

It says the answer should be f'(x) = -2sin2x.

chudzoik said:
It says the answer should be f'(x) = -2sin2x.

Recall this identity: $2sin(x)cos(x) = sin(2x)$

Oh wow forgot entirely about that. These kinds of mistakes will be the end of me. Thanks for the help.

## What are the basic differentiation rules for sinusoidal functions?

The basic differentiation rules for sinusoidal functions are as follows:
1. The derivative of a sine function is equal to the cosine function.
2. The derivative of a cosine function is equal to the negative sine function.
3. The derivative of a tangent function is equal to the secant squared function.
4. The derivative of a cotangent function is equal to the negative cosecant squared function.
5. The derivative of a secant function is equal to the secant function multiplied by the tangent function.

## How do you differentiate a product of a sine and cosine function?

To differentiate a product of a sine and cosine function, you can use the product rule, which states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function. In the case of a sine and cosine function, this would result in the derivative being equal to the negative sine function multiplied by the cosine function, plus the cosine function multiplied by the sine function.

## What is the chain rule and how is it used to differentiate a composite sinusoidal function?

The chain rule is a rule for finding the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. To use the chain rule to differentiate a composite sinusoidal function, you would first identify the outer and inner functions, and then apply the chain rule accordingly.

## Can the quotient rule be used to differentiate a fraction with a sine or cosine function in the numerator or denominator?

Yes, the quotient rule can be used to differentiate a fraction with a sine or cosine function in the numerator or denominator. The quotient rule states that the derivative of a fraction is equal to the denominator multiplied by the derivative of the numerator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator. This rule can be applied to fractions with sinusoidal functions by treating the sine or cosine functions as the numerator or denominator respectively.

## What is the relationship between the differentiation rules for sine and cosine functions?

The differentiation rules for sine and cosine functions are closely related. This is because the derivative of a sine function is equal to the cosine function, and the derivative of a cosine function is equal to the negative sine function. This relationship is known as the derivative of a sine function being the negative cosine function, and the derivative of a cosine function being the negative sine function. It is important to remember this relationship when differentiating trigonometric functions.

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