What is the derivative of #arctan(x/3)#?

2 Answers
Jim G.
Aug 22, 2017

#1/(3+x^2/3)#

Explanation:

#"differentiate using the "color(blue)"chain rule"#

#•color(white)(x)d/dx(arctan(f(x)))=1/(1+(f(x))^2)xxf'(x)#

#rArrd/dx(arctan(x/3))#

#=1/(1+x^2/9)xxd/dx(1/3x)#

#=1/(3(1+x^2/9))=1/(3+x^2/3)#

Jim H
Aug 22, 2017

If you haven't memorized the drivative of #arctan(x)#. (or you don't trust your memory), see below.

Explanation:

#y = arctan(x/3)#

#tany = x/3#

Diiferentiate implicitly.

#sec^2 y dy/dx = 1/3#

#dy/dx = 1/3 cos^2(y)#

Use #tany = x/3# to find #cos y = 3/sqrt(x^2+9)# (See Note below)

#dy/dx = 1/3 (3/sqrt(x^2+9))^2#

#dy/dx = 3/(x^2+9)#

Note
There are several possible methods to do this.
I like to draw a right triangle with one angle labeled #y#. The side opposite #y# is #x# and the adjacent side is #3#, so ythe hypotenuse is #sqty(x^2+9)# and the cosine is #3/sqrt(x^2+9)#
Others prefer to use #tan^2 y + 1 = sec^2 y# to find #sec^2y# and the invert.

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