Find the derivative if y=sec^-1 (x+3)
To find the derivative of the given function y = sec^(-1)(x+3), we can use the chain rule.
The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of f(g(x)) with respect to x is given by the product of the derivative of f with respect to g times the derivative of g with respect to x.
Let's break down the function step by step:
Step 1: Identify the outer function and the inner function.
In our case, the outer function is sec^(-1)(x) and the inner function is x + 3.
Step 2: Differentiate the outer function.
To find the derivative of sec^(-1)(x), we can rewrite it using an identity:
sec^(-1)(x) = arccos(1 / x)
Now we can differentiate arccos(1 / x) using the chain rule:
d/dx[arccos(1 / x)] = -1 / (sqrt(1 - (1 / x)^2)) * d/dx(1 / x)
Step 3: Differentiate the inner function.
To find the derivative of x + 3, we differentiate it as a linear function:
d/dx(x + 3) = 1
Step 4: Apply the chain rule.
Since the derivative of the outer function is -1 / (sqrt(1 - (1 / x)^2)) * d/dx(1 / x), and the derivative of the inner function is 1, we can apply the chain rule:
dy/dx = (-1 / (sqrt(1 - (1 / (x + 3))^2))) * 1
Simplifying the expression, we get:
dy/dx = -1 / sqrt(1 - (1 / (x + 3))^2)
Therefore, the derivative of y = sec^(-1)(x + 3) is dy/dx = -1 / sqrt(1 - (1 / (x + 3))^2).