Math Problem Statement

Find the derivative f'(π/4) where f(x) = (6x + cos(2x)) / (1 + sin^2(2x))

Solution

We are tasked with finding the derivative f(π4)f' \left( \frac{\pi}{4} \right), where the function f(x)f(x) is given by:

f(x)=6x+cos(2x)1+sin2(2x)f(x) = \frac{6x + \cos(2x)}{1 + \sin^2(2x)}

We will use the quotient rule to differentiate this function. The quotient rule states that if f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}, then the derivative f(x)f'(x) is:

f(x)=g(x)h(x)g(x)h(x)[h(x)]2f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}

Here, g(x)=6x+cos(2x)g(x) = 6x + \cos(2x) and h(x)=1+sin2(2x)h(x) = 1 + \sin^2(2x).

Step 1: Differentiate g(x)g(x)

We have: g(x)=6x+cos(2x)g(x) = 6x + \cos(2x) Differentiating term by term: g(x)=62sin(2x)g'(x) = 6 - 2\sin(2x)

Step 2: Differentiate h(x)h(x)

We have: h(x)=1+sin2(2x)h(x) = 1 + \sin^2(2x) Using the chain rule, we get: h(x)=2sin(2x)cos(2x)2=4sin(2x)cos(2x)h'(x) = 2 \cdot \sin(2x) \cdot \cos(2x) \cdot 2 = 4\sin(2x)\cos(2x) Using the double angle identity, 2sin(2x)cos(2x)=sin(4x)2\sin(2x)\cos(2x) = \sin(4x), we simplify: h(x)=2sin(4x)h'(x) = 2\sin(4x)

Step 3: Apply the Quotient Rule

Now, applying the quotient rule: f(x)=(62sin(2x))(1+sin2(2x))(6x+cos(2x))2sin(4x)(1+sin2(2x))2f'(x) = \frac{(6 - 2\sin(2x))(1 + \sin^2(2x)) - (6x + \cos(2x)) \cdot 2\sin(4x)}{(1 + \sin^2(2x))^2}

Step 4: Evaluate at x=π4x = \frac{\pi}{4}

We now substitute x=π4x = \frac{\pi}{4} into the expressions.

For g(x)g'(x): g(π4)=62sin(π2)=62=4g' \left( \frac{\pi}{4} \right) = 6 - 2\sin \left( \frac{\pi}{2} \right) = 6 - 2 = 4

For h(x)h(x): h(π4)=1+sin2(π2)=1+12=2h \left( \frac{\pi}{4} \right) = 1 + \sin^2 \left( \frac{\pi}{2} \right) = 1 + 1^2 = 2

For h(x)h'(x): h(π4)=2sin(4π4)=2sin(π)=0h' \left( \frac{\pi}{4} \right) = 2\sin(4 \cdot \frac{\pi}{4}) = 2\sin(\pi) = 0

Thus, the derivative at x=π4x = \frac{\pi}{4} is: f(π4)=42(6π4+cos(π2))022=84=2f' \left( \frac{\pi}{4} \right) = \frac{4 \cdot 2 - (6 \cdot \frac{\pi}{4} + \cos \left( \frac{\pi}{2} \right)) \cdot 0}{2^2} = \frac{8}{4} = 2

So, the correct answer is 2\boxed{2}.

Would you like more details on any of the steps?

Here are 5 related questions:

  1. How do you use the chain rule in trigonometric functions?
  2. What is the significance of the quotient rule in calculus?
  3. How do trigonometric identities simplify derivative calculations?
  4. How would this calculation change if the function were altered slightly, like cos(2x)\cos(2x) replaced by sin(2x)\sin(2x)?
  5. How can you interpret the result geometrically at x=π4x = \frac{\pi}{4}?

Tip: Always simplify the derivative expression as much as possible before plugging in specific values to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Trigonometric Functions

Formulas

Quotient rule: f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2
Chain rule for derivatives
Double angle identity: 2sin(2x)cos(2x) = sin(4x)

Theorems

Quotient Rule
Chain Rule
Trigonometric Identities

Suitable Grade Level

College level calculus