Math Problem Statement
Find yn, if y = tan^−1 (x/c)
Solution
To find (the nth derivative of ) if , let's follow these steps.
Step 1: Initial function
We are given:
Step 2: First derivative
The derivative of with respect to is: Now, apply the chain rule. Let , then: Since , we get: Simplifying:
Step 3: Second derivative
Now, find the second derivative by differentiating : Using the quotient rule, , where:
- , so
- , so
Applying the quotient rule:
Step 4: Higher derivatives
We can see a pattern emerging in the derivatives. The first derivative is positive, and the second derivative introduces a negative factor with in the numerator. We can generalize that:
- Odd derivatives will have terms involving raised to odd powers and will alternate in sign.
- Even derivatives will have terms involving raised to even powers, also alternating in sign.
For higher-order derivatives , the pattern involves successively differentiating using the quotient rule. Each differentiation introduces higher powers of in the numerator and increases the power of in the denominator.
Would you like to continue with more specific higher-order derivatives or explore this pattern in more depth?
Related Questions:
- How is the inverse tangent function derived?
- What is the geometric interpretation of ?
- Can we use implicit differentiation to find higher derivatives of inverse trigonometric functions?
- How does the behavior of change as ?
- What are some applications of inverse trigonometric functions in calculus?
Tip:
When differentiating functions involving inverse trigonometric expressions, always check for simplifications to reduce the complexity of higher derivatives.
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Math Problem Analysis
Mathematical Concepts
Calculus
Inverse Trigonometric Functions
Differentiation
Formulas
y = tan^−1(x/c)
dy/dx = 1 / (1 + u^2)
Quotient rule: (f'(x)g(x) - f(x)g'(x)) / g(x)^2
Theorems
Chain Rule
Quotient Rule
Suitable Grade Level
College level (Calculus I/II)