Math Problem Statement

Find the derivative of the function g(x) = sin(6 - 7x).

Solution

To find the derivative of the function g(x)=sin(67x)g(x) = \sin(6 - 7x), we apply the chain rule.

  1. First, the derivative of sin(u)\sin(u) is cos(u)\cos(u), where u=67xu = 6 - 7x.
  2. Then, we take the derivative of the inner function 67x6 - 7x, which is 7-7.

Thus, the derivative of g(x)g(x) is:

g(x)=cos(67x)(7)=7cos(67x).g'(x) = \cos(6 - 7x) \cdot (-7) = -7\cos(6 - 7x).

Would you like more details on how the chain rule was applied?

Here are five related questions you might find useful:

  1. What is the chain rule, and how is it applied in calculus?
  2. How do you differentiate trigonometric functions involving composite functions?
  3. How would the derivative change if the function was g(x)=cos(67x)g(x) = \cos(6 - 7x)?
  4. How do you differentiate a function involving multiple trigonometric terms?
  5. Can you apply the chain rule to other non-trigonometric functions, like exponentials?

Tip: When differentiating composite functions, always start by identifying the "outer" function and apply the chain rule to account for the derivative of the "inner" function.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Trigonometry
Chain Rule

Formulas

d/dx[sin(u)] = cos(u) * du/dx
d/dx[6 - 7x] = -7

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12