How do you differentiate #f(x)=e^tan(1/x^2) # using the chain rule?

1 Answer
Dec 23, 2015

Answer:

Use substitution

Explanation:

#f(x) = e^(tan(1/x^2))#
Assume #t=1/x^2# then #p = tan(t)# then
#f(p) = e^p#
#(df)/(dx) = (df)/(dp) * (dp)/(dt) *(dt)/(dx)#
# = e^p*sec^2t *(-2/x^3)#
Now back substitute

# = e^(tan(1/x^2)) \times sec^2(1/x^2) \times ((-2)/x^3)#

Another approach

Take #ln# on both sides
#ln(f(x)) = tan(1/x^2)#

Now differentiate both sides

#1/f \times (df)/(dx) = sec^2(1/x^2) \times ((-2)/x^3)#
# (df)/(dx) = f \times sec^2(1/x^2) \times ((-2)/x^3)#
# = e^(tan(1/x^2)) \times sec^2(1/x^2) \times ((-2)/x^3)#