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If #f(x) =xe^x# and #g(x) = e^(3x)#, what is #f'(g(x)) #?

Answer 1

Charles Arocha

(e^{3x}+1)e^{e^{3x}}

# f'(x) = \frac{d}{dx} [ xe^x ] #

use product rule: # \frac{d}{dx} [ xe^x ] = 1[e^x] + x[e^x] # # = (x+1)e^x #

plug in # x = e^{3x} # into # f'(x) = (x+1)e^x # :

# f'(g(x)) = (e^{3x}+1)e^{e^{3x}} #

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Answer 2

Axel Alvarez

To find ( f'(g(x)) ), first find ( g'(x) ), then substitute ( g(x) ) into ( f'(x) ), and finally evaluate the expression.

Given: [ f(x) = x e^x ] [ g(x) = e^{3x} ]

[ g'(x) = 3 e^{3x} ]

[ f'(x) = e^x + x e^x ]

[ f'(g(x)) = e^{3x} + e^{3x} \cdot 3x ]

[ f'(g(x)) = e^{3x} + 3x e^{3x} ]

Therefore, ( f'(g(x)) = e^{3x} + 3x e^{3x} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.