Math Problem Statement
Solution
We are asked to find the equation of the tangent line to the graph of the function:
at the point .
Step 1: Find the derivative of
To get the equation of the tangent line, we need to compute the derivative , which gives us the slope of the tangent line at a given point.
Given , we will apply the chain rule for differentiation:
- The outer function is , where .
- The derivative of is .
Now, we need to find the derivative of . Using the power rule, the derivative is:
Now, applying the chain rule:
Step 2: Compute the slope at
To find the slope of the tangent line at , we substitute into the derivative :
Since , we simplify the expression:
We know that because has a period of and corresponds to an odd multiple of . Therefore:
So, the slope of the tangent line at is .
Step 3: Equation of the tangent line
The general form of the equation of a line is:
where is the slope, and is the point of tangency. In our case, , and the slope . Substituting these values:
Simplifying:
Thus, the equation of the tangent line is:
Would you like more details on any part of the solution?
Here are some related questions:
- How is the chain rule used in differentiation?
- What is the geometric significance of a tangent line?
- How do you find the cosine of an angle like ?
- Why is the derivative important for finding the slope of a tangent line?
- How does the power rule apply to square root functions?
Tip: When differentiating composite functions, always identify the inner and outer functions clearly to apply the chain rule properly.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Tangent Line
Chain Rule
Formulas
f'(x) = cos(9√x) * (9 / (2√x))
Equation of tangent line: y - y1 = m(x - x1)
Theorems
Chain Rule in Differentiation
Suitable Grade Level
Grades 11-12
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