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- GIVEN THE FUNCTION F OF X
WE WANT TO DETERMINE THE DERIVATIVE FUNCTION
AND THEN FIND THE VALUE OF THE DERIVATIVE FUNCTION AT X = 1
AND THE EQUATION OF THE TANGENT LINE AT X = 1.
AFTER WE DO ALL THIS,
WE WILL VERIFY IT GRAPHICALLY AS WELL.
FOR THE FIRST STEP WE WANT TO FIND THE DERIVATIVE FUNCTION
GIVEN F OF X = -2X SQUARED + 4.
SO F PRIME OF X IS GOING TO BE EQUAL TO -2
x YOUR DERIVATIVE OF X SQUARED,
SO WE'LL APPLY THE POWER RULE GIVEN HERE.
SO WE'RE GOING TO MULTIPLY BY THE EXPONENT
AND THEN SUBTRACT 1 FROM THE EXPONENT,
2 - 1 = 1 + THE DERIVATIVE OF 4.
THE DERIVATIVE OF A CONSTANT = 0, SO WE HAVE + 0.
SO OUR DERIVATIVE FUNCTION F PRIME OF X = -4X.
SO THIS IS THE FIRST PART OF OUR ANSWER.
NEXT WE'RE ASKED THE FIND THE DERIVATIVE FUNCTION VALUE
WHEN X = 1 WHICH MEANS WE WANT TO FIND F PRIME OF 1,
SO WE'LL SUBSTITUTE 1 FOR X IN OUR DERIVATIVE FUNCTION.
THAT'S GOING TO BE -4 x 1. SO F PRIME OF 1 = -4.
NOW THIS DERIVATIVE FUNCTION VALUE REPRESENTS
THE SLOPE OF THE TANGENT LINE AT X = 1.
SO FOR THE LAST QUESTION,
WHEN WE'RE ASKED TO FIND THE EQUATION
OF THE TANGENT LINE AT X = 1,
WE KNOW THE SLOPE OF OUR TANGENT LINE IS GOING TO BE -4.
BUT IN ORDER TO FIND THE EQUATION OF THE TANGENT LINE
WE ALSO NEED TO FIND A POINT ON THE LINE.
WELL, WE KNOW THE X-COORDINATE IS GOING TO BE 1,
AND THE Y-COORDINATE AT THE POINT WOULD BE F OF 1.
SO NOW WE NEED TO FIND F OF 1
TO FIND THE Y-COORDINATE OF THE POINT ON OUR TANGENT LINE.
WELL, F OF 1 WOULD JUST BE -2 x 1 SQUARED + 4.
WELL, THIS WOULD BE -2 x 1 IS -2 + 4,
SO THE FUNCTION VALUE WITH THE Y-COORDINATE IS +2.
SO NOW WE KNOW THE SLOPE OF OUR TANGENT LINE IS -4,
AND THE LINE ALSO CONTAINS THE POINT (1,2).
AND NOW TO FIND THE EQUATION OF OUR TANGENT LINE
WE'LL GO AHEAD AND USE
THE POINT-SLOPE FORM OF A LINE GIVEN HERE.
SO WE'LL HAVE Y - Y1 WHICH IS 2 = M
WHICH IS (-4 x X) - (X SUB 1) WHICH IS +1.
SO NOW WE'LL GO AHEAD AND CLEAR THESE PARENTHESES
AND SOLVE FOR Y.
SO WE'LL DISTRIBUTE HERE, AND THEN WE'LL ADD 2.
SO WE HAVE Y - 2 = THIS WILL BE -4X + 4.
ADD 2 TO BOTH SIDES, AND WE HAVE Y = -4X + 6
WHICH WOULD BE THE EQUATION
OF OUR TANGENT LINE AT THE POINT (1,2).
LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY.
THE BLUE GRAPH IS THE GRAPH OF OUR ORIGINAL FUNCTION.
THIS POINT T IS OUR POINT OF TANGENCY
WHICH HAD THE COORDINATES (1,2),
AND THIS RED LINE IS OUR TANGENT LINE
THAT HAS AN EQUATION Y = -4X + 6.
I HOPE YOU FOUND THIS HELPFUL.