How do you differentiate #y=sec^-1(x^7)#?
2 Answers
Explanation:
differentiate wrt
we now substitute back to express the derivative as a function of
# dy/dx= 7/(xsqrt(x^14 -1)) # , or#7/(x^8sqrt(1-1/x^14)) #
Explanation:
Let
# y = sec^(-1)(x^7) #
Then:
# sec y = x^7#
Differentiating Implicitly:
# sec y tany dy/dx= 7x^6#
# :. x^7 tany dy/dx= 7x^6#
# :. tany dy/dx= 7x^6/x^7#
# :. tany dy/dx= 7/x#
And using the identity
# tan^2 y = sec^2y -1 #
# " " = (x^7)^2 -1 #
# " " = x^14 -1 #
# :. tan y =sqrt(x^14 -1) #
Substituting we get:
# sqrt(x^14 -1) dy/dx= 7/x #
# :. dy/dx= 7/(xsqrt(x^14 -1)) #
Which can also be written:
# dy/dx = 7/(xsqrt(x^14(1-1/x^14))) #
# " " = 7/(x*x^7*sqrt(1-1/x^14)) #
# " " = 7/(x^8sqrt(1-1/x^14)) #