What is the equation of the tangent line of f(x)=(x+3)(x−1)ex at x=-3?

Expert answers to "What is the equation of the tangent line..."

plastikffxi117h

2022-03-23

What is the equation of the tangent line of

f (x) = \frac{(x + 3) (x - 1)}{e^{x}}

at x=-3?

umgebautv6v2

Beginner2022-03-24Added 10 answers

Explanation:
First to find the slope of the tangent, find derivative of f(x):

f^{'} (x) = \frac{d}{d x} (x + 3) (x - 1) \cdot e^{- x}

= \frac{d}{d x} [(x^{2} + 2 x - 3) \cdot e^{- x}]

we will differentiate using u.v form and chain rule.

u = x^{2} + 2 x - 3

du=2x+2

v = e^{- x}

d v = - e^{- x}

f^{'} (x) = x^{2} + 2 x - 3 \cdot (- e^{- x}) + e^{- x} \cdot (2 x + 2)

Now we want the particular slope at x = -3. So lets substitute x as -3 in f '(x)

f^{'} (- 3) = (9 - 9) \cdot - e^{3} + e^{3} \cdot - 4

so equation of tangent at x = -3 is
y=mx=c

\Rightarrow y = (- 4 e^{3}) x + c

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