Math Problem Statement

differentiate. 4sec(x) - csc(x)

Solution

To differentiate the function f(x)=4sec(x)csc(x)f(x) = 4 \sec(x) - \csc(x), we will use the known derivatives of sec(x)\sec(x) and csc(x)\csc(x):

  • The derivative of sec(x)\sec(x) is sec(x)tan(x)\sec(x) \tan(x).
  • The derivative of csc(x)\csc(x) is csc(x)cot(x)-\csc(x) \cot(x).

Now, apply these rules to the given function:

f(x)=ddx(4sec(x)csc(x))f'(x) = \frac{d}{dx} \left( 4 \sec(x) - \csc(x) \right)

f(x)=4sec(x)tan(x)(csc(x)cot(x))f'(x) = 4 \cdot \sec(x) \tan(x) - (-\csc(x) \cot(x))

f(x)=4sec(x)tan(x)+csc(x)cot(x)f'(x) = 4 \sec(x) \tan(x) + \csc(x) \cot(x)

Thus, the derivative is:

f(x)=4sec(x)tan(x)+csc(x)cot(x)f'(x) = 4 \sec(x) \tan(x) + \csc(x) \cot(x)

Let me know if you'd like further details or have any questions! Here are 5 related questions you can explore:

  1. How do the derivatives of trigonometric functions like secant and cosecant differ from those of sine and cosine?
  2. Can you explain why the derivative of sec(x)\sec(x) is sec(x)tan(x)\sec(x) \tan(x)?
  3. How would you find the second derivative of this function f(x)f(x)?
  4. What are some applications of the derivatives of trigonometric functions in real-world problems?
  5. How does the chain rule apply when differentiating functions involving trigonometric identities?

Tip: When differentiating trigonometric functions, always recall the standard derivatives of functions like sine, cosine, secant, and cosecant, as they form the foundation of most such problems!

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Math Problem Analysis

Mathematical Concepts

Differentiation
Trigonometric Functions
Secant
Cosecant

Formulas

Derivative of sec(x) is sec(x)tan(x)
Derivative of csc(x) is -csc(x)cot(x)

Theorems

Rules for Differentiating Trigonometric Functions

Suitable Grade Level

Grades 11-12