How do you differentiate #sqrt(cos(x^2+2))+sqrt(cos^2x+2)#?

Question

Charles Anctil · Accepted Answer

#(dy)/(dx)= (xsen(x^2+2)+sen(x+2))/(sqrtcos(x^2+2)+sqrt(cos^2(x+2)))# #(dy)/(dx)= 1/(2sqrtcos(x^2+2)+sqrt(cos^2(x+2))) *sen(x^2+2)*2x+2sen(x+2)# #(dy)/(dx)= (2xsen(x^2+2)+2sen(x+2))/(2sqrtcos(x^2+2)+sqrt(cos^2(x+2)))# #(dy)/(dx)= (cancel2(xsen(x^2+2)+sen(x+2)))/(cancel2sqrtcos(x^2+2)+sqrt(cos^2(x+2)))# #(dy)/(dx)= (xsen(x^2+2)+sen(x+2))/(sqrtcos(x^2+2)+sqrt(cos^2(x+2)))#

Isaiah Allen · Answer

undefined To differentiate the expression sqrt(cos(x^2+2)) + sqrt(cos^2x + 2), you can use the chain rule. The derivative of sqrt(u) with respect to x is (1/2)*u^(-1/2)*du/dx. Applying this rule to each term, the derivatives are as follows:

For the first term: 
Derivative of sqrt(cos(x^2+2)) = (1/2)*(cos(x^2+2))^(-1/2) * derivative of (cos(x^2+2)) with respect to x.

For the second term:
Derivative of sqrt(cos^2x + 2) = (1/2)*(cos^2x + 2)^(-1/2) * derivative of (cos^2x + 2) with respect to x.

To find the derivatives of the inner functions, you need to apply the chain rule and the derivative of cos(x) which is -sin(x). After finding the derivatives, simplify the expressions.