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alexparker
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Prove that (d/dx)arcos(x) = -1/(Sqrt(1-x^2)
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SUMMARY
The derivative of the inverse cosine function, arccos(x), is proven to be -1/(Sqrt(1-x^2)). This is established by differentiating the identity cos(arccos(x)) = x. By letting y = arccos(x) and applying implicit differentiation, the relationship dx/dy = -sin(y) leads to dy/dx = -1/sin(y). Utilizing trigonometric identities and right triangle properties further simplifies the expression to the final derivative.
PREREQUISITES- Understanding of inverse trigonometric functions
- Knowledge of implicit differentiation techniques
- Familiarity with trigonometric identities
- Basic concepts of right triangle geometry
- Study implicit differentiation in calculus
- Learn about trigonometric identities and their applications
- Explore the properties of inverse trigonometric functions
- Practice problems involving derivatives of inverse functions
Students studying calculus, particularly those focusing on differentiation of inverse trigonometric functions, as well as educators seeking to clarify these concepts for their students.