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

1 Answer
Jan 17, 2016

Answer:

Use #d/dx(e^u) = e^u (du)/dx# and #d/dx(sqrtu) = 1/(2sqrtu) (du)/dx# or rewrite the function first.

Explanation:

Using the chain rule to differentiate the function as presented:

#f'(x) = e^sqrt(1/x^2)d/dx(sqrt(1/x^2))#

# = e^sqrt(1/x^2) * 1/(2sqrt(1/x^2)) d/dx(1/x^2)#

# = e^sqrt(1/x^2) * 1/(2sqrt(1/x^2)) ((-2)/x^3)#

# = (-sqrt(x^2) * e^sqrt(1/x^2))/x^3#

Rewriting the function as a piecewise function first

Since #sqrtx^2 = absx = {(-x,"," , x < 0),(x,",", 0 < x) :}#, we can rewrite the function as:

#f(x) = {(e^(-1/x),"," , x < 0),(e^(1/x) ,",", 0 < x) :}#

Differentiating each part gets us:

#f'(x) = {(e^(1/x)/x^2,"," , x < 0),(-e^(1/x)/x^2 ,",", 0 < x) :}#

This is equivalent to the answer given above.