How do you differentiate # f(x)=e^((ln(x^2+3)^2)# using the chain rule.?

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Answer 1 · 2016-04-09T20:44:39

#f'(x) = 4x^3 + 12x#

The equation is in a ridiculous form above. You can cancel out the #e# and the power #ln#, leaving just

#e^(ln(x^2 + 3)^2) = (x^2 + 3)^2#

The chain rule says that if

#f(x) = g(h(x))#

then

#f'(x) = h'(x) * g'(h)#.

Substituting #h(x) = x^2 + 3# and #g(h) = h^2#, then

#h'(x) = 2x#

#g'(h) = 2h = 2(x^2 + 3) = 2x^2 + 6#

Multiplying these together according to the chain rule,

#h'(x) * g'(h) = 2x * (2x^2 + 6) = 4x^3 + 12x#

which is the final answer.

If you want to work it out another way, expand the brackets to get #x^4 + 6x^2 + 9# and differentiate with the usual method.