How do you find the derivative of the function: #(arccos(x/6))^2#?

1 Answer
Jan 23, 2017

Let #h(x) = x/6#
Let #g(h) = arccos(h)#
Let #f(g) = g^2#
To differentiate, use the chain rule with a function nested within a function within a function.

Explanation:

The chain rule with a function nested within a function within a function:

#(d((arccos(x/6)^2)))/dx = (d(f(g(h(x)))))/dx#

#(d((arccos(x/6)^2)))/dx = (df)/(dg)(dg)/(dh)(dh)/dx" [1]"#

#(df)/(dg) = 2g = 2arccos(h) = 2arccos(x/6)#

#(dg)/(dh) = -1/sqrt(1 - h^2) = -1/sqrt(1 - (x/6)^2)#

#(dh)/dx = 1/6#

Substituting into equation [1]

#(d((arccos(x/6)^2)))/dx = (2arccos(x/6))(-1/sqrt(1 - (x/6)^2))(1/6)#

#(d((arccos(x/6)^2)))/dx = (-2arccos(x/6))/(6sqrt(1 - (x/6)^2))#

#(d((arccos(x/6)^2)))/dx = (-2arccos(x/6))/sqrt(36 - x^2)#