find the derivative of the function. y=3tan^-1(x+sqrt1+x^2)
To find the derivative of the function y = 3tan^(-1)(x + sqrt(1+x^2)), we can use the chain rule.
Let u = x + sqrt(1+x^2).
Then, we have y = 3tan^(-1)(u).
To find dy/dx, we can find dy/du and du/dx separately.
1. Finding dy/du:
Using the derivative of arctan function, we have:
dy/du = 3 * 1/(1+u^2).
Substituting u = x + sqrt(1+x^2), we have:
dy/du = 3 * 1/(1+(x+sqrt(1+x^2))^2).
2. Finding du/dx:
Taking the derivative of u = x + sqrt(1+x^2), we have:
du/dx = 1 + (1/2)(2x)(1+x^2)^(-1/2).
Simplifying, we get:
du/dx = 1 + x/(sqrt(1+x^2)).
Now, using the chain rule, we can multiply dy/du and du/dx to find dy/dx:
dy/dx = (dy/du) * (du/dx)
= (3/(1+(x+sqrt(1+x^2))^2)) * (1 + x/(sqrt(1+x^2)))
= 3(1 + x/(sqrt(1+x^2))) / (1+(x+sqrt(1+x^2))^2)
Therefore, the derivative of the function y = 3tan^(-1)(x + sqrt(1+x^2)) is:
dy/dx = 3(1 + x/(sqrt(1+x^2))) / (1+(x+sqrt(1+x^2))^2)