d/dx arcsec(x)
To find the derivative of the inverse secant function, we can use the formula for the derivative of an inverse function:
If y = f^(-1)(x), then the derivative of y with respect to x is given by dy/dx = 1 / (dx/dy)
To find the derivative of the inverse secant function, we can use the formula for the derivative of an inverse function:
If y = f^(-1)(x), then the derivative of y with respect to x is given by dy/dx = 1 / (dx/dy).
In this case, let y = arcsec(x). This means that x = sec(y).
We know that the derivative of secant function is given by d/dx sec(x) = sec(x) tan(x). So, we can find dx/dy:
dx/dy = d/dy sec(y).
Using the chain rule, we have:
dx/dy = d/dy [sec(x)] = sec(y) tan(y).
Now, we can find dy/dx using the formula for the derivative of an inverse function:
dy/dx = 1 / (dx/dy) = 1 / (sec(y) tan(y)).
But we know that x = sec(y), so we can substitute sec^(-1)(x) for y in the equation:
dy/dx = 1 / (sec(sec^(-1)(x)) tan(sec^(-1)(x))).
Now, we can simplify further:
sec(sec^(-1)(x)) = x, since sec^(-1)(x) is the angle whose secant is x.
tan(sec^(-1)(x)) = √(sec^2(sec^(-1)(x)) – 1) = √(x^2 – 1), using the Pythagorean identity for tangent.
Therefore, dy/dx = 1 / (x √(x^2 – 1)).
So, the derivative of arcsec(x) with respect to x is 1 / (x √(x^2 – 1)).
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