Math Problem Statement
Solve 21 and 16
Solution
Let's solve problems 16 and 21:
Problem 16
Given the function: We are asked to find the derivative and evaluate the slope of the tangent line at .
Step 1: Find the derivative .
This is a quotient, so we use the quotient rule: For our function:
- and
- and
Applying the quotient rule: Simplify:
Step 2: Evaluate the derivative at .
The slope of the tangent line at is .
Problem 21
Given the function: We are asked to find at .
Step 1: Find the derivative .
Rewrite the function:
Use the power rule and the chain rule: Simplify:
Step 2: Evaluate the derivative at .
The value of at is .
Do you want more details or have any questions?
Here are some related questions to expand your understanding:
- How would the slope change if we evaluated problem 16 at a different point?
- What are the key steps for applying the quotient rule?
- How does the chain rule help in problem 21?
- What would happen to the derivative if the functions in problems 16 and 21 were squared?
- How do we determine if a tangent line is horizontal?
Tip: Remember that the quotient rule requires careful application of the numerator and denominator derivatives—keeping track of signs is crucial!
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Math Problem Analysis
Mathematical Concepts
Differentiation
Quotient Rule
Power Rule
Chain Rule
Formulas
Quotient rule: (d/dx)(u/v) = (v(du/dx) - u(dv/dx)) / v^2
Power rule: d/dx (x^n) = n*x^(n-1)
Chain rule: d/dx (f(g(x))) = f'(g(x)) * g'(x)
Theorems
Derivative properties
Tangent line slope
Suitable Grade Level
Grades 11-12
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