DMac said:
thomas49th said:
DMac said:
The formula for differentiating ln(secx) is d/dx[ln(secx)] = secx*tanx.
When x is between -pi/2 and 0, the formula for differentiating ln(secx) is d/dx[ln(secx)] = secx*tanx. However, you must also use the chain rule to account for the inner function, which is secx. Therefore, the final answer is d/dx[ln(secx)] = secx*tanx*secx = sec^2x*tanx.
Yes, for example, if x = -pi/4, the formula for differentiating ln(secx) becomes d/dx[ln(secx)] = sec(-pi/4)*tan(-pi/4) = -sqrt(2)*(-1) = sqrt(2).
To evaluate the derivative of ln(secx) at a specific value of x, simply plug in the value of x into the derivative formula. For example, if x = -pi/3, the derivative would be d/dx[ln(sec(-pi/3))] = sec(-pi/3)*tan(-pi/3) = -2*sqrt(3).
The given range of -pi/2 <= x <= 0 is important because it ensures that the function ln(secx) is well-defined and differentiable. This is because the range excludes any values of x that would make the secant function undefined, such as x = 0 or x = pi/2, which would result in a division by zero. Therefore, this range allows us to differentiate ln(secx) without encountering any undefined values.