perplexus.info :: Calculus : Given Functional Equation, Find the Derivative

Given Functional Equation, Find the Derivative (Posted on 2024-06-18)

Given that:
2*f(sin x) + f(cos x) = x

Find f'(x)

No Solution Yet Submitted by K Sengupta

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Solution

Comment 2 of 2 |

Lets start with a trick: substitute x -> pi/2-x.

This simplifies to 2*f(cos x) + f(sin x) = pi/2 - x.

Next up take a linear combination of the two equations 2*[2*f(sin x) + f(cos x) = x] - [2*f(cos x) + f(sin x) = pi/2 - x].

This yields f(sin x) = x - pi/6.

Then a simple substitution of x -> arcsin(x) gives us f(x) = arcsin(x) - pi/6. Then take the derivative to get f'(x) = 1/sqrt[1-x^2].

Posted by Brian Smith on 2024-06-19 10:37:09

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