Differentiating Inverse Trigonometric Functions
Differentiating Inverse Trigonometric Functions
Understanding Inverse Trigonometric Functions
- Get to know more about inverse trigonometric functions, also known as arc functions (arcsin x, arccos x, arctan x).
- Understand that these are the inverse functions of the basic trigonometric functions - sine, cosine, and tangent. They allow you to calculate the angle when given its sine, cosine, or tangent.
- Remember that these functions have specific domains. For example, the domain of arcsin and arccos is
-1 ≤ x ≤ 1and the domain of arctan is the set of all real numbers.
Differentiation of Arcsin and Arccos Functions
- Know how to compute the derivatives of arcsin and arccos with the help of basic differentiation rules.
- Understand that the derivatives of arcsin and arccos functions are represented by the formulae:
d/dx(arcsin x) = 1/sqrt(1 - x^2)andd/dx(arccos x) = -1/sqrt(1 - x^2)
Differentiation of Arctan Function
- Learn how to calculate the derivative of an arctan function.
- Understand that the derivative is given by:
d/dx(arctan x) = 1/(1 + x^2)
Differentiating More Complex Inverse Trigonometric Functions
- Understand how to apply chain rule to differentiate more complex inverse trigonometric functions. This involves differentiating a function within another function.
- Remember that the chain rule formula,
d/dx f(g(x)) = f'(g(x)) * g'(x), essentially applies the derivative of the outer function first and then of the inner function.
Practice and Implementation
- Ensure to practice using these formulas to become familiar with differentiating inverse trigonometric functions.
- Try to solve progressively more complex problems to test your understanding.
- Utilize these principles to solve real-world physics and geometry problems where application of inverse trigonometric functions may be necessary.