What is the derivative of #-(5-x^2)^(1/2) #?

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Answer 1 · 2017-11-05T09:35:55

#x / sqrt(5-x^2)#

#d/dx (-sqrt(5-x^2))# Refer to this as Expression [ A ]

We will pull the negative sign ( - ) out, and rewrite as

#- d/dx (sqrt(5-x^2))# Refer to this as Expression [ B ]

We will use the Chain Rule now to move on, as there are two functions to deal with

Chain Rule states that #dy/dx = (dy/(du)) * ((du)/dx)#

In Expression [ A ], let #u = 5 - x^2#

We will rewrite the Expression [ B ] as

#- d/ (du) (sqrt(u)) * d/(dx) (5 - x ^ 2)# Refer to this as Expression [ C ]

We can write #d /(du) (sqrt(u))# as # 1/(2*sqrt(u))#

Similarly, we can write #d /(dx) (5 - x^2)# as #( -2*x)#

Using these two results in Expression [ C ], we obtain

#-1/(2 * sqrt(u)) * (-2*x)#

Substitute back #u = (5 - x^2)#

#-(1/(2 sqrt(5-x^2))) * (-2*x)#

This expression simplifies to

#(2*x)/(2*sqrt(5-x^2)#

This expression simplifies to

#x/(sqrt(5 - x^2)# is our final answer.