Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. Understanding how to find and evaluate inverse functions is crucial for solving problems involving derivatives of inverses.
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Derivative of Inverse Functions
The derivative of an inverse function can be calculated using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship highlights how the rate of change of the inverse function at a point is the reciprocal of the rate of change of the original function at the corresponding point.
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Derivatives of Inverse Sine & Inverse Cosine
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is often applied when dealing with inverse functions, as it helps in understanding how changes in one variable affect another.
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