What is the derivative of #tan^7(x^2)#?
1 Answer
Aug 14, 2017
Explanation:
We're asked to find the derivative
#d/(dx) [tan^7(x^2)]#
We can first use the chain rule:
#d/(dx) [tan^7(x^2)] = d/(du) [u^7] (du)/(dx)#
where
-
#u = tan(x^2)# -
#d/(du) [u^7] = 7u^6# :
#= 7d/(dx)[tan(x^2)]tan^6(x^2)#
Using the chain rule again:
#d/(dx) [tan(x^2)] = d/(du) [tanu] (du)/(dx)#
where
-
#u = x^2# -
#d/(du) [tanu] = sec^2u# :
#= 7d/(dx)[x^2]sec^2(x^2)tan^6(x^2)#
Use the power rule on the
#d/(dx) [x^n] = nx^(n-1)#
where
#n = 2# :
#= 7(2x)sec^2(x^2)tan^6(x^2)#
#color(blue)(ulbar(|stackrel(" ")(" "= 14xsec^2(x^2)tan^6(x^2)" ")|)#
