Problem

For the function f(x)=x2+2, find the equation of the tangent line at x=−2.

Answer

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Answer

y=−4x−2

Steps

Step 1 ：First, find the slope of the tangent line by taking the derivative of the function: f′(x)=2x

Step 2 ：Next, find the slope at the point x=−2: f′(−2)=2(−2)=−4

Step 3 ：Now, find the y-coordinate of the point on the function: f(−2)=(−2)2+2=4+2=6

Step 4 ：The point on the function is (−2,6) and the slope of the tangent line is −4

Step 5 ：Use the point-slope form of a line: y−y1=m(x−x1)

Step 6 ：Plug in the point and slope: y−6=−4(x+2)

Step 7 ：Simplify the equation: y−6=−4x−8

Step 8 ：Add 6 to both sides: y=−4x−2

Step 9 ：y=−4x−2

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