Math Problem Statement
Solution
The problem in the image is as follows:
Problem: The function has a tangent at that forms a angle with the -axis and passes through the point .
Additionally, the function is defined as:
You are tasked to find the derivative of at , i.e., .
Solution:
Step 1: Analyze the tangent line
The slope of a line forming a angle with the -axis is:
So, at , the derivative of , , is:
The tangent line passes through the point . Using the point-slope form of a line equation, the tangent line can be expressed as: Since , this becomes: At , the tangent line crosses the -axis, so:
Thus:
Step 2: Differentiate
The function is given as:
To find , use the product rule: Applying the product rule:
The derivative of involves the chain rule:
Thus:
Step 3: Evaluate
Substitute into :
Simplify:
From Step 1, we know:
Substitute these values:
Final Answer:
Would you like me to elaborate on any part of the solution?
5 Related Questions:
- How do we compute derivatives using the chain rule in general cases?
- What is the product rule for differentiation, and when is it applied?
- How does the slope of a tangent line relate to the derivative of a function?
- How can we verify the slope of a line given two points?
- What is the importance of defining composite functions like ?
Tip: Always ensure that the slope of a tangent is consistent with the angle provided, as small errors here can lead to incorrect solutions.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Chain Rule
Product Rule
Tangent Line
Formulas
Product Rule: (uv)' = u'v + uv'
Chain Rule: f(g(x))' = f'(g(x)) * g'(x)
Slope of a line forming 45° with x-axis: m = tan(45°) = 1
Theorems
Derivative rules
Point-slope form of a line equation
Suitable Grade Level
Grades 11-12 or college level
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