`f(x) = (x - 4)(x^2 + 6x - 1), (0,4)` Find an equation of the tangent line to the graph of `f` at the given point.

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Chapter 2, Review - Problem 42 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=(x-4)(x^2+6x-1),(0, 4)`

Find the derivative of the function using the Product Rule. Then plug in the given x value into the derivative function to calculate the slope.

`f'(x)=(x-4)(2x+6)+(x^2+6x-1)(1)`

`f'(0)=(0-4)(2(0)+6)+(0^2+6(0)-1)`

` ` `f'(0)=(-4)(6)+(-1)`

`f'(0)=-24-1=-25`

Use the slope -25 and the given point (0, 4) to find the equation of the tangent line at the specified point.

`y-4=-25(x-0)`

`y=-25x+4`

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