What is the derivative of? : #y=3tan^(-1)(x+sqrt(1+x^2) ) #
1 Answer
Oct 19, 2017
# dy/dx = ( 3(1+x/(sqrt(1+x^2))))/(1+(x+sqrt(1+x^2))^2) #
Explanation:
We seek (I assume):
# dy/dx # where#y=3tan^(-1)(x+sqrt(1+x^2) ) #
We will need the standard result:
# d/dx tan^(-1)x = 1/(1+x^2) #
And the power rule, in conjunction with the chain rule, for:
# d/dx (x+ sqrt(1+x^2) ) = 1+d/dx (1+x^2)^(1/2)#
# " " = 1+1/2(1+x^2)^(-1/2) d/dx (1+x^2)#
# " " = 1+1/(2sqrt(1+x^2)) (2x)#
# " " = 1+x/(sqrt(1+x^2))#
Then we can apply the product rule to the original function to get:
# dy/dx = 3 {1/(1+(x+sqrt(1+x^2))^2) } d/dx (x+sqrt(1+x^2) ) #
# \ \ \ \ \ = 3 {1/(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) #
