Get the solution to "What is the equation of the tangent line..."

Max Grant

Answered question

2022-03-16

What is the equation of the tangent line of

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

at x=3?

Urijah Burch

Beginner2022-03-17Added 4 answers

Finding f'(3) will tell us the slope of the tangent line when x=3. To figure out the point on the function, find f(3).

f (3) = 3 e^{3} - 6

The point will be

(3, 3 e^{3} - 6)

.
Finding the derivative will require the product rule. Also remember that

\frac{d}{d x} [e^{x}] = e^{x}

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

f^{'} (x) = e^{x} + x e^{x} - 2 x + 1

f^{'} (3) = e^{3} + 3 e^{3} - 6 + 1 = 4 e^{3} - 5

Write in point-slope form:

y - 3 e^{3} + 6 = (4 e^{3} - 5) (x - 3)

In slope-intercept form:

y = (4 e^{3} - 5) x - 9 e^{3} + 9

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