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nlexa [21]
2 years ago
11

Finding second Derivatives In Exercise,find the second derivate. f(x) = (3 + 2x)e - 3x

Mathematics
1 answer:
Talja [164]2 years ago
7 0

Answer:

the question is incomplete, the complete question is "Finding second Derivatives In Exercise,find the second derivate.

f(x)=(3+2x)e^{-3x}"

answer:\frac{df(x)^{2}}{dx^{2}}=(15+18x)e^{-3x}\\,

Step-by-step explanation:

To determine the second derivative, we differentiate twice.

for the first differentiation, we use the product rule approach. i.e

f(x)=u(x)v(x)\\\frac{df(x)}{dx}=u(x)\frac{dv(x)}{dx}+ v(x)\frac{du(x)}{dx}\\

from f(x)=(3+2x)e^{-3x} if w assign

u(x)=(3+2x)  and the derivative, \frac{du(x)}{dx}=2

also v(x)=e^{-3x} and the derivative  \frac{dv(x)}{dx}=-3e^{-3x}.

If we substitute values we arrive at

\frac{df(x)}{dx}=(3+2x)(-3e^{-3x})+2e^{-3x}\\\frac{df(x)}{dx}=(-9-6x)e^{-3x}+2e^{-3x}\\\frac{df(x)}{dx}=(-9-6x+2)e^{-3x}\\\frac{df(x)}{dx}=-(7+6x)e^{-3x}\\,

Now to determine the second derivative we use the product rule again

this time, u(x)=(7+6x)  and the derivative, \frac{du(x)}{dx}=6

also v(x)=e^{-3x} and the derivative  \frac{dv(x)}{dx}=-3e^{-3x}.

If we substitute values we arrive at

\frac{df(x)^{2}}{dx^{2}}=-((7+6x)(-3e^{-3x})+6e^{-3x})\\\frac{df(x)^{2}}{dx^{2}}=-((-21-18x)e^{-3x}+6e^{-3x})\\\frac{df(x)^{2}}{dx^{2}}=-((-21-18x+6)e^{-3x})\\\frac{df(x)^{2}}{dx^{2}}=(15+18x)e^{-3x}\\,

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