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Lesson 6 Definition of Derivatives and the Four Step Rule

# Lesson 6 Definition of Derivatives and the Four Step Rule ...

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THE DERIVATIVE OBJECTIVES: to define the derivative of a function

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Derivative of a Function The process of finding the derivative of a function is called differentiation and the branch of calculus that deals with this process is called differential calculus . Differentiation is an important mathematical tool in physics, mechanics, economics
Consider a point on the curve )) ( , ( 2 2 x f x Q ), ( x f y = that is distinct from and compute the slope of the secant line through P and Q. )), ( , ( 1 1 x f x P PQ m x x f x f m PQ - = ) ( ) ( 1 2 where 1 2 x x x - = x x x + = 1 2 and x x f x x f m PQ - + = ) ( ) ( 1 1

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If we let approach then the point Q will move along the curve and approach point P. As point Q approaches P, the value of approaches zero and the secant line through P and Q approaches a limiting position, then we will consider that position to be the position of the tangent line at P. 2 x , 1 x x
)) ( , ( 1 1 x f x P )) ( , ( 2 2 x f x Q ) ( x f y = x x x x x x + = - = 1 2 1 2 y tangent line secant line x y

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