Find the derivative if y=sec^-1(x+3)
To find the derivative of y = sec^(-1)(x+3), we can use the chain rule.
First, let's rewrite the function y = sec^(-1)(x+3) using inverse trigonometric identities.
Recall that sec^(-1)(x) is equivalent to cos^(-1)(1/x) or arccos(1/x).
So, we have y = arccos(1/(x+3)).
Now, let's find the derivative.
Using the chain rule, the derivative of y with respect to x can be found by differentiating the outer function (arccos) and then multiplying it by the derivative of the inner function (1/(x+3)).
The derivative of arccos(u), where u = 1/(x+3), is given by -1/sqrt(1-u^2).
So, the derivative of y = arccos(1/(x+3)) is
dy/dx = (-1)/sqrt(1-(1/(x+3))^2).
Simplifying further, we have
dy/dx = (-1)/sqrt(1-(1/(x+3))^2).
This is the derivative of y = sec^(-1)(x+3).