Taylor Series (Derivative question)

In summary, the question is about taking the derivative of f(x)=cos^2(x) and simplifying it to -sin(2x). The solution involves using well-known trig identities such as sin(2x)=2sin(x)cos(x). The person is also seeking help for their upcoming final exam.
  • #1
newbe318
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I was looking at the solution for problem 6 and I am confused on taking the derivatives of the function f(x)= cos^2 (x)
I took the first derivative and did get the answer f^(1) (x)= 2(cos(x)) (-sin (x)), but how does that simplify to -sin (2x)?

Is there some trig identity that I am not aware of?
Your help would be much appreciated, since I have a final Friday.
Thank you.
 
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  • #2
newbe318 said:
View attachment 93145

I was looking at the solution for problem 6 and I am confused on taking the derivatives of the function f(x)= cos^2 (x)
I took the first derivative and did get the answer f^(1) (x)= 2(cos(x)) (-sin (x)), but how does that simplify to -sin (2x)?

Is there some trig identity that I am not aware of?
This is a well-known trig identity.
##\sin(2x) = 2\sin(x)\cos(x)##
Another is ##\cos(2x) = \cos^2(x) - \sin^2(x)##
The right side can be written in two other forms: ##2\cos^2(x) - 1## or ##1 - 2\sin^2(x)##.
newbe318 said:
Your help would be much appreciated, since I have a final Friday.
Thank you.
 
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Likes jim mcnamara

FAQ: Taylor Series (Derivative question)

Q1: What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms, each term being a polynomial function of the variable. It is a way to approximate a function by using its derivatives at a single point.

Q2: What is the purpose of a Taylor series?

The purpose of a Taylor series is to approximate a function and its derivatives at a specific point. This can be useful in many applications, such as solving differential equations, estimating values of a function, and understanding the behavior of a function.

Q3: What is the formula for a Taylor series?

The formula for a Taylor series is: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

Q4: How do you find the coefficients of a Taylor series?

The coefficients of a Taylor series can be found by taking derivatives of the function at the given point and plugging them into the formula. The nth coefficient is equal to the nth derivative of the function evaluated at the point, divided by n factorial.

Q5: What is the relationship between Taylor series and derivatives?

Taylor series and derivatives are closely related. The Taylor series is a representation of a function using its derivatives, and the derivatives of a function can be used to find the coefficients of the Taylor series. Taylor series can also be used to approximate the value of a function at a specific point by using its derivatives.

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