What is the derivative of this function #arc cot(x/5)#?

1 Answer
Steve M · mason m
Nov 9, 2016

# d/dx arc cot(x/5 )= -5/(25+x^2) #

Explanation:

We can write #y=arc cot(x/5) <=>coty=x/5 #

We can then differentiate implicitly:

# -csc^2y dy/dx = 1 /5#
# dy/dx= -1/(5csc^2y) #

Using the identity # 1+cot^2A=csc^2A # we have

# dy/dx = -1/(5(1+(x/5 )^2)) #
# dy/dx = -1/(5(1+x^2/25 )) * 25/25 #
# dy/dx = -(25)/(5(25+x^2)) #
# dy/dx = -5/(25+x^2) #

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