Applications of the Derivative: The Power Rule


[ MUSIC ] – WELCOME TO APPLICATIONS USING
DIFFERENTIATION TECHNIQUES. WE’LL COVER THE CONSTANT RULE,
POWER RULE, CONSTANT MULTIPLE RULE, AND THE SUM OF DIFFERENCE RULE
WITH AN APPLICATIONS. THE GOALS WILL BE TO FIND
THE EQUATION OF A TANGENT LINE AND FIND THE POINTS
ON A FUNCTION THAT HAVE A HORIZONTAL
TANGENT LINE. LET’S TAKE A LOOK AT AN EXAMPLE. FIND THE EQUATION OF THE TANGENT
LINE AT THE POINT (4,14) TO THE FUNCTION Y=X SQUARED
– THE SQUARE ROOT OF X. ON OUR PREVIOUS VIDEO
WE DISCUSSED THAT WE CAN’T USE THE POWER RULE
WHEN IT’S IN RADICAL FORM. SO THE FIRST STEP WILL BE TO
REWRITE THIS RATIONAL EXPONENTS. SO WE’LL HAVE Y=X SQUARED – X
TO THE POWER OF 1/2. NOW, IF WE WANT TO FIND THE
EQUATION OF THE TANGENT LINE, OF COURSE WE’RE GOING TO HAVE
TO FIND THE SLOPE OF THAT LINE. SO IF WE FIND THE DERIVATIVE
AND EVALUATE IT AT X=4, WE’LL FIND THE SLOPE
OF THAT TANGENT LINE. LET’S TRY IT. Y PRIME IS EQUAL
TO THE DERIVATIVE OF X SQUARED, WHICH WILL BE 2X –
THE DERIVATIVE OF X TO THE 1/2. THAT WOULD BE 1/2 x X
TO THE POWER OF 1/2 – 1, WHICH WOULD BE -1/2. NOW WE’LL SIMPLIFY THIS. SIMPLIFYING THIS TERM, IF I MOVE THIS DENOMINATOR
I’LL HAVE X TO THE +1/2 POWER, WHICH IS THE SAME
AS THE SQUARE ROOT OF X. NOW I NEED TO EVALUATE THIS
WHEN X IS 4. SO WE’D HAVE 2 x 4 – 1 DIVIDED
BY 2 x THE SQUARE ROOT OF 4. SO Y PRIME OF 4 WILL BE EQUAL
TO 8 – SQUARE ROOT OF 4 IS 2 x THIS 2 WOULD GIVE US A – 1/4
OR – 1/4, WHICH WOULD EQUAL 31/4 OR 7.75. NOW, THIS WILL BE THE SLOPE
OF OUR TANGENT LINE. SO NOW WE NEED TO FIND
THE EQUATION OF THE TANGENT LINE AND WE’LL USE POINT SLOPE FORM. JUST IN CASE YOU DON’T REMEMBER
THAT, HERE IT IS. AGAIN, OUR SLOPE IS HERE,
OUR POINT IS HERE. THIS WOULD GIVE US Y – 14
IS EQUAL TO 7.75 x THE QUANTITY X – 4. WE’LL SOLVE THIS FOR Y. ADD 14 TO BOTH SIDES AND WE HAVE
Y IS EQUAL TO 7.75X – 17. OKAY. LET’S VERIFY OUR WORK
BY LOOKING AT A GRAPH. HERE WE HAVE OUR FUNCTION. HERE WE HAVE THE GRAPH
OF OUR TANGENT LINE. HERE’S THE POINT THAT WAS
REFERRED TO IN THE PROBLEM, (4,14). AND WE CAN SEE THAT OUR LINE
IS TANGENT AT THAT POINT, WHICH VERIFIES OUR WORK. LET’S TAKE A LOOK
AT ANOTHER EXAMPLE. FIND THE POINTS ON THE FUNCTION FOR WHICH THE TANGENT LINE
WOULD BE HORIZONTAL. WELL, ALL HORIZONTAL LINES
HAVE A SLOPE OF ZERO, SO ON THIS PROBLEM
WE NEED TO FIND WHERE THE DERIVATIVE WOULD BE
EQUAL TO ZERO. REMEMBER THE DERIVATIVE GIVES US
THE SLOPE OF THE TANGENT LINES. SO LET’S FIND OUR DERIVATIVE
AND SET IT EQUAL TO ZERO. SO DERIVATIVE OF -1/3X CUBE
WOULD BE -1/3 x THE DERIVATIVE OF X CUBE, WHICH WOULD BE 3X SQUARED, + 6 x THE DERIVATIVE
OF X SQUARED THAT WOULD BE 2X. – 11 X THE DERIVATIVE OF X
THAT WOULD BE 1, AND THEN – DERIVATIVE OF 50
THAT WOULD BE 0. LET’S SIMPLIFY HERE
AND SET IT EQUAL TO ZERO. THOSE THREE SIMPLIFY. THIS WOULD LEAVE US
WITH -X SQUARED + 12X – 11. AGAIN, WE WANT TO KNOW
WHEN THIS IS EQUAL TO ZERO. THAT’S WHERE THE TANGENT LINES
WOULD BE HORIZONTAL. LET’S FACTOR OUT THE NEGATIVE. NOTICE IT CHANGES THE SIGN
OF ALL OF THE TERMS. AND THIS DOES FACTOR. THE FACTORS OF +11 AT THE -12
THAT’D BE -11 AND A -1. SO THE SOLUTIONS TO THIS
WOULD BE X=11 AND X=1. NOW, THESE ARE
THE X COORDINATES, BUT THEY ASK FOR THE POINTS, SO WE ALSO HAVE TO FIND
THE Y COORDINATES. SO WE NEED TO FIND F OF 11. THAT WOULD GIVE US
THE Y COORDINATE WHEN X IS 11 AND F OF 1. REMEMBER WE’RE GOING TO SUB
THESE VALUES BACK INTO THE ORIGINAL FUNCTION
HERE NOT THE DERIVATIVE. SAVE TIME,
I’VE ALREADY DONE THAT. F OF 11 IS EQUAL TO 111.3
REPEATING. F OF 1 IS EQUAL TO -55.3
REPEATING. SO THE COORDINATES OF THE POINTS WHERE THE TANGENT LINES
WOULD BE HORIZONTAL ARE (11,111.3 REPEATING),
AND (1,-55.3 REPEATING). AGAIN, LET’S VERIFY IT
WITH A GRAPH. IN BLUE WE HAVE OUR FUNCTION. IN RED YOU CAN SEE OUR
TWO HORIZONTAL TANGENT LINES. THE POINTS OF TANGENCY
ARE THE POINTS THAT WE FOUND HERE AND HERE. I WOULD ALSO LIKE YOU TO NOTICE
THE RELATIVE EXTREMA OCCUR WHERE THE DERIVATIVE
IS EQUAL TO ZERO. HERE WE HAVE A RELATIVE MINIMUM, AND HERE WE HAVE
A RELATIVE MAXIMUM. WE’LL COME BACK TO THIS IDEA. THANK YOU FOR WATCHING.
I HOPE THAT HELPS. HAVE A WONDERFUL DAY.  

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