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)" ")|)