Math Problem Statement

Defreantiation of combosit function cos5x

Solution

To differentiate the composite function cos(5x)\cos(5x), we again apply the chain rule.

Given f(x)=cos(5x)f(x) = \cos(5x), where g(x)=5xg(x) = 5x, the chain rule tells us:

ddx[cos(g(x))]=sin(g(x))g(x)\frac{d}{dx} [\cos(g(x))] = -\sin(g(x)) \cdot g'(x)

For the given function, g(x)=5xg(x) = 5x, so:

g(x)=ddx[5x]=5g'(x) = \frac{d}{dx} [5x] = 5

Thus, the derivative of cos(5x)\cos(5x) is:

ddx[cos(5x)]=sin(5x)5=5sin(5x)\frac{d}{dx} [\cos(5x)] = -\sin(5x) \cdot 5 = -5\sin(5x)

So, the derivative of cos(5x)\cos(5x) is 5sin(5x)-5\sin(5x).

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

  1. How do you differentiate cos(2x)\cos(2x)?
  2. What is the derivative of sin(5x)\sin(5x)?
  3. How do you apply the chain rule to functions like cos(ax+b)\cos(ax + b)?
  4. What is the derivative of cos1(x)\cos^{-1}(x)?
  5. How does the product rule work when combined with trigonometric functions?
  6. What is the second derivative of cos(5x)\cos(5x)?
  7. Can you derive the derivative using trigonometric identities?
  8. How would you differentiate cos2(5x)\cos^2(5x)?

Tip: When using the chain rule, always differentiate the outer function first and then multiply by the derivative of the inner function.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Chain Rule
Trigonometric Functions

Formulas

Chain rule for differentiation

Theorems

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Suitable Grade Level

Advanced High School